A noncommutative probability theory
S. P.
Gudder;
R. L.
Hudson
1-41
Abstract: A noncommutative probability theory is developed in which no boundedness, finiteness, or ``tracial'' conditions are imposed. The underlying structure of the theory is a ``probability algebra'' $(\mathcal{a},\omega )$ where $\mathcal{a}$ is a *-algebra and $\omega$ is a faithful state on $\mathcal{a}$. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.
Explosions in completely unstable flows. I. Preventing explosions
Zbigniew
Nitecki
43-61
Abstract: Several conditions are equivalent to the property that a flow (on an open manifold) and its ${C^0}$ perturbations have only wandering points. These conditions are: (i) there exists a strong Liapunov function; (ii) there are no generalized recurrent points in the sense of Auslander; (iii) there are no chain recurrent points, in the sense of Conley; (iv) there exists a fine sequence of filtrations; (v) relative to some metric; the flow is the gradient flow of a function without critical points. We establish these equivalences, and consider a few questions related to structural stability when all orbits wander.
Explosions in completely unstable flows. II. Some examples
Zbigniew
Nitecki
63-88
Abstract: A dynamical system with all points wandering is ``explosive'' if some $ {C^0}$ perturbation has nonwandering points. It is known that the plane admits no explosive cascades or flows; in this paper, examples are constructed to show that all open manifolds except ${R^1}$ and ${R^2}$ admit explosive flows (and hence cascades).
A $q$-analog of restricted growth functions, Dobinski's equality, and Charlier polynomials
Stephen C.
Milne
89-118
Abstract: We apply finite operator techniques due to G. C. Rota to a combinatorial identity, which counts a collection of generalized restricted growth functions in two ways, and obtain a q-analog of Charlier polynomials and Dobinski's equality for the number of partitions of an n-set. Our methods afford a unified proof of certain identities in the combinatorics of finite dimensional vector spaces over $ {\text{GF}}(q)$.
Conformality and semiconformality of a function holomorphic in the disk
Shinji
Yamashita
119-138
Abstract: Conformality and semiconformality at a boundary point, of a function f nonconstant and holomorphic in $\left\vert z \right\vert < 1$ are local properties. Therefore one would suspect the requirement of such global conditions on f as f is univalent in $ \left\vert z \right\vert < 1$, or f is a member of a larger class which contains all univalent functions in $\left\vert z \right\vert < 1$. We shall prove some extensions and new results without any assumption on f, or with a local assumption on f at most. Our methods are, for the most part, different from the ones in the classical cases. One of the main tools is Theorem 8 on the angular limits of the real part of a holomorphic function and its derivative.
Unitary invariance in algebraic algebras
Charles
Lanski
139-146
Abstract: A structure theorem is obtained for subspaces invariant under conjugation by the unitary group of a prime algebraic algebra over an infinite field. For an invariant subalgebra W, it is shown that either W is central, W contains an ideal, or the ring satisfies the standard identity of degree eight. Also, for prime algebras not satisfying such an identity, the unitary group is not solvable.
Symmetric duality for structured convex programs
L.
McLinden
147-181
Abstract: A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar's general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.
Dolbeault homotopy theory
Joseph
Neisendorfer;
Laurence
Taylor
183-210
Abstract: For complex manifolds, we define ``complex homotopy groups'' in terms of the Dolbeault complex. Many theorems of classical homotopy theory are reflected in the properties of complex homotopy groups. Analytic fibre bundles yield long exact sequences of complex homotopy groups and various Hurewicz theorems relate complex homotopy groups to the Dolbeault cohomology. In a more analytic vein, the classical Fröhlicher spectral sequence has a complex homotopy analogue. We compute these complex homotopy invariants for such examples as Calabi-Eckmann manifolds, Stein manifolds, and complete intersections.
The real and rational cohomology of differential fibre bundles
Joel
Wolf
211-220
Abstract: Consider a differential fibre bundle (E, $\pi$, X, ${G \mathord{\left/ {\vphantom {G H}} \right. \kern-\nulldelimiterspace} H}$ , G). Under certain reasonable hypotheses, the cohomology of the total space E is computed in terms of the cohomology of the base space X and algebraic invariants of the imbedding of H into G.
Projective modules for finite Chevalley groups
John W.
Ballard
221-249
Abstract: The purpose of this paper is to obtain character formulas for certain indecomposable projective modules for a finite Chevalley group. It is shown that these modules are also modules for the corresponding semisimple algebraic group.
Hyperbolicity and cycles
J. E.
Franke;
J. F.
Selgrade
251-262
Abstract: In this paper cycle points are defined without the assumption of Axiom A. The closure of the set of cycle points $\mathcal{C}$ being quasi-hyperbolic is shown to be equivalent to Axiom A plus no cycles. Also we give a sufficient condition for $ \mathcal{C}$ to equal the chain recurrent set. In proving these theorems, a spectral decomposition for quasi-hyperbolic invariant sets is used.
Stability of isometries
Peter M.
Gruber
263-277
Abstract: A map $ T:E \to F$ (E, F Banach spaces) is called an $\varepsilon$-isometry if $ \left\vert\,{\left\Vert {T(x)-T(y)} \right\Vert-\left\Vert{x -y}\right\Vert}\,\right\vert\, \leqslant \varepsilon$ whenever $x,\,y \in E$. Hyers and Ulam raised the problem whether there exists a constant $\kappa$, depending only on E, F, such that for every surjective $\varepsilon$-isometry $T:E \to F$ there exists an isometry $I:E \to F$ with ${\left\Vert {T(x) - I(x)} \right\Vert}\leqslant \kappa \varepsilon$ for every $x \in E$. It is shown that, whenever this problem has a solution for E, F, one can assume $\kappa \leqslant 5$. In particular this holds true in the finite dimensional case.
The rigidity of graphs
L.
Asimow;
B.
Roth
279-289
Abstract: We regard a graph G as a set $ \{ 1, \ldots , v \}$ together with a nonempty set E of two-element subsets of $ \{ 1, \ldots , v \}$. Let $p = ({p_1},\ldots,{p_v})$ be an element of $ {\textbf{R}^{nv}}$ representing v points in ${\textbf{R}^n}$. Consider the figure $G(p)$ in $ {\textbf{R}^n}$ consisting of the line segments $ [{p_i},{p_j}]$ in ${\textbf{R}^n}$ for $ \{ i,j\} \in E$. The figure $G(p)$ is said to be rigid in ${\textbf{R}^n}$ if every continuous path in ${\textbf{R}^{nv}}$, beginning at p and preserving the edge lengths of $G(p)$, terminates at a point $q \in {\textbf{R}^{nv}}$ which is the image $(T{p_1}, \ldots ,T{p_v})$ of p under an isometry T of ${\textbf{R}^n}$. Otherwise, $G(p)$ is flexible in ${\textbf{R}^n}$. Our main result establishes a formula for determining whether $G(p)$ is rigid in ${\textbf{R}^n}$ for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in $ {\textbf{R}^3}$, and other related matters.
Hamiltonian systems in a neighborhood of a saddle point
Viorel
Barbu
291-307
Abstract: The behavior of Hamiltonian differential systems associated with a concave convex function H in a Hilbert space is studied by variational methods. It is shown that under quite general conditions on the function H the system behaves in a neighborhood of a minimax saddle point of H much like as in the classical theory of ordinary differential systems. The results extend previous work of R. T. Rockafellar.
Chern classes of certain representations of symmetric groups
Leonard
Evens;
Daniel S.
Kahn
309-330
Abstract: A formula is derived for the Chern classes of the representation id $ \int {\xi :P\int {H \to {U_{pn}}} }$ where P is cyclic of order P and $\xi :H \to {U_n}$ is a fintie dimensional unitary representation of the group H. The formula is applied to the problem of calculating the Chern classes of the ``natural'' representations ${\pi _j}:{\mathcal{S}_j} \to {U_j}$ of symmetric groups by permutation matrices.
An integral Riemann-Roch formula for induced representations of finite groups
Leonard
Evens;
Daniel S.
Kahn
331-347
Abstract: Let H be a subgroup of the finite group G, $\xi$ a finite dimensional complex representation of H and $\rho$ the induced representation of G. If ${s_k}(\rho ) \in {H^{2k}}(G,\textbf{Z})$, $k \geqslant 1$ denote the characteristic classes bearing the same relation to power sums that Chern classes bear to elementary symmetric functions, then we prove the following, $\displaystyle \bar N (k)( {{s_k}(\rho ) - {\text{T}}{{\text{r}}_{H \to G}}({s_k}(\xi ))}) = 0,$ (1) where $\displaystyle \bar N(k) = {\prod _{\begin{array}{*{20}{c}} {p\vert N(k)} {p{\text{prime}}} \end{array}}}p$ (2) and $\displaystyle N(k) = \left( {\begin{array}{*{20}{c}} {\prod\limits_{p{\text{prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right)/k!.$ (3) (Tr denotes transfer.) Moreover, $\bar N (k)$ is the least integer with this property. This settles a question originally raised in a paper of Knopfmacher in which it was conjectured that the required bound was N(k).
Quantitative Korovkin theorems for positive linear operators on $L\sb{p}$-spaces
H.
Berens;
R.
DeVore
349-361
Abstract: Let $({L_n})$ be a sequence of positive linear operators on $ {L_p}(\Omega )$, $1 \leqslant p < \infty$ or $C(\Omega )$ with $\Omega \subseteq {R^m}$. For suitable $\Omega$, the functions $({\varphi _i})_{i = 0}^{m + 1}$ given by ${\varphi _0}(x) \equiv 1$, ${\varphi _i}(x) \equiv {x_i}$, $1 \leqslant i \leqslant m$,and $ {\varphi _{m + 1}}(x) \equiv {\left\vert x \right\vert^2}$ form a test set for $ {L_p}(\Omega )$. That is, if ${L_n}({\varphi _i})$ converges to ${\varphi _i}$ in ${\left\Vert \cdot \right\Vert _p}$ for each $i = 0,1, \ldots ,m + 1$, then $ {L_n}(f)$ converges to f in $ {\left\Vert \cdot \right\Vert _p}$ for each $ f \in {L_p}(\Omega )$. We give here quantitative versions of this result. Namely, we estimate ${\left\Vert {f - {L_n}f} \right\Vert _p}$ in terms of the error $ {\left\Vert {{\varphi _i} - {L_n}{\varphi _i}} \right\Vert _p}$, $0 \leqslant i \leqslant m + 1$,and the smoothness of the function f.
On complete hypersurfaces of nonnegative sectional curvatures and constant $m$th mean curvature
Philip
Hartman
363-374
Abstract: The main result is that if $M = {M^n}$ is a complete Riemann manifold of nonnegative sectional curvature and $X:\,M \to {R^{n + 1}}$ is an isometric immersion such that $X(M)$ has a positive constant mth mean curvature, then $X(M)$ is the product of a Euclidean space ${R^{n - d}}$ and a d-dimensional sphere, $m \leqslant d \leqslant n$.
Harmonic functions and mass cancellation
J. R.
Baxter
375-384
Abstract: If a function on an open set in $ {\textbf{R}^n}$ has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius $\delta (x)$, a function of x. Under appropriate conditions on $\delta$ and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function $ \delta$ is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.
On the algebraic criteria for local Pareto optima. II
Yieh Hei
Wan
385-397
Abstract: Constrained vector optimization problems in the large are studied in this paper. Fixing any constraints so that the feasible set is a manifold with corners, we prove that the set of local Pareto optima for typical vector-valued functions admit Whitney prestratifications. Furthermore, these prestratifications persist under small perturbation of vector-valued functions. The main tools used here are a variant of Thom's transversality theorem and the stratification theory of semialgebraic sets.
Measurable parametrizations and selections
Douglas
Cenzer;
R. Daniel
Mauldin
399-408
Abstract: Let W be a Borel subset of $I \times I$ (where $I = [0,1]$) such that, for each x, $ {W_x} = \{ y:\,(x,y) \in W\}$ is uncountable. It is shown that there is a map, g, of $I \times I$ onto W such that (1) for each x, $ g(x, \cdot )$ is a Borel isomorphism of I onto ${W_x}$ and (2) both g and ${g^{ - 1}}$ are $ S(I \times I)$-measurable maps. Here, if X is a topological space, $ S(X)$ is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that $S(X)$ is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone. This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann. We also show an analogous result holds if W is only assumed to be analytic.
Analytic equations and singularities of plane curves
John J.
Wavrik
409-417
Abstract: Theorems (Artin, Wavrik) exist which show that sufficiently good approximate (power series) solutions to a system of analytic equations may be approximated by convergent solutions. This paper considers the problem of explicity determining the order, $\beta$, to which an approximate solution must solve the system of equations. The paper deals with the case of one equation, $ f(x,y) = 0$, in two variables. It is shown how $\beta$ depends on the singularities of the curve $f(x,y) = 0$. A method for obtaining the minimal $ \beta$ is given. A rapid way of finding $\beta$ using the Newton Polygon for f applies in special cases.
A remark on zeta functions
Jun-ichi
Igusa
419-429
Abstract: In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as ``test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin's conjecture is true.
Semifree actions on finite groups on homotopy spheres
John
Ewing
431-442
Abstract: We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial.
A unified approach to measurable and continuous selections
G.
Mägerl
443-452
Abstract: An abstract selection theorem is presented which contains as special cases-among others-the measurable selection theorem of Kuratowski and Ryll-Nardzewski, as well as the continuous selection theorem of Michael.
$4$-manifolds, $3$-fold covering spaces and ribbons
José María
Montesinos
453-467
Abstract: It is proved that a PL, orientable 4-manifold with a handle presentation composed by 0-, 1-, and 2-handles is an irregular 3-fold covering space of the 4-ball, branched over a 2-manifold of ribbon type. A representation of closed, orientable 4-manifolds, in terms of these 2-manifolds, is given. The structure of 2-fold cyclic, and 3-fold irregular covering spaces branched over ribbon discs is studied and new exotic involutions on $ {S^4}$ are obtained. Closed, orientable 4-manifolds with the 2-handles attached along a strongly invertible link are shown to be 2-fold cyclic branched covering spaces of ${S^4}$. The conjecture that each closed, orientable 4-manifold is a 4-fold irregular covering space of ${S^4}$ branched over a 2-manifold is reduced to studying $ \gamma \char93 {S^1} \times {S^2}$ as a nonstandard 4-fold irregular branched covering of ${S^3}$.
Group actions on $A\sb{k}$-manifolds
Hsü Tung
Ku;
Mei Chin
Ku
469-492
Abstract: By an $ {A_k}$-manifold we mean a connected manifold with elements $ {w_i} \in {H^1}(M),\,1 \leqslant i \leqslant k$, such that ${w_1} \cup \, \cdots \cup \,{w_k} \ne 0$. In this paper we study the fixed point set, degree of symmetry, semisimple degree of symmetry and gaps of transformation groups on ${A_k}$-manifolds.
A new approach to the limit theory of recurrent Markov chains
K. B.
Athreya;
P.
Ney
493-501
Abstract: Let $\{ {X_n};\,n \geqslant 0\} $ be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times $\{ {N_i};\,i \geqslant 1\} $ such that $ \{ {X_{{N_i}}};{\text{ }}i \geqslant 1\}$ are independent and identically distributed. This idea is used to show that $\{ {X_n}\}$ is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.
Contractible $3$-manifolds of finite genus at infinity
E. M.
Brown
503-514
Abstract: A class of contractible open 3-manifolds is defined. It is shown that all contractible open 3-manifolds which can be written as a union of cubes with a bounded number of handles are in this class. It is shown that a proper map between manifolds of this class which induces an isomorphism of proper fundamental groups (e.g. a proper homotopy equivalence) is proper homotopic to a homeomorphism. A naturality condition for homomorphisms of proper fundamental groups is developed. It is shown that a natural homomorphism between the proper fundamental groups of these manifolds is induced by a proper map.