Metric entropies of sets in abstract Wiener space

Let (X,H,μ) be an abstract Wiener space, E(?,K) denote the metric entropy of a set K⊂X. If K is not a slim set, then we prove that

     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

The cycle space of an embedded graph

Let G be a connected graph with edge set E embedded in the surface ∑. Let G° denote the geometric dual of G. For a subset d of E, let τd denote the edges of G° that are dual to those edges of G in d. We prove the following generalizations of well-known facts about graphs embedded in the plane. (1) b is a boundary cycle in G if and only if τb is a cocycle in G°. (2) If T is a spanning tree of G, then τ(E/T) contains a spanning tree of G°. (3) Let T be any spanning tree of G and, for e ? E/T, let T(e) denote the fundamental cycle of e. Let U ∪ E/T. Then τU is a spanning tree of G° if and only if the set of face boundaries, less any one, together with the set {T(e); e ? E/(T ∪ U)} is a basis for the cycle space of G.     

A Banach-Dieudonné theorem for germs of holomorphic functions

Let $\text{H}$(U) denote the space of all holomorphic functions on an open subset U of a complex Fréchet space E. Let $\text{H}$(K) denote the space of all holomorphic germs on a compact subset K of E. It is shown that $\text{H}$(K), with a natural topology, is the inductive limit of a suitable sequence of compact subsets, within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by Nachbin coincides with the compact-open topology on $\text{H}$(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of P. Boland and S. Dineen [Bull. Soc. Math. France106 (1978), 311–336], R. Meise [Proc. Roy. Irish Acad. Sect. A81 (1981), 217–223], and others.     

An invariant measure for the loop space of a simply connected compact symmetric space

Let X denote a simply connected compact Riemannian symmetric space, U the universal covering of the identity component of the group of automorphisms of X, and LU the loop group of U. In this paper we prove the existence (and conjecture the uniqueness) of an LU-invariant probability measure on a distributional completion of the loop space of X.     

Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation

In this paper, we first consider the least-squares solution of the matrix inverse problem as follows： Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ｜｜AX - B｜｜. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is： Given an arbitrary A^＊, find a matrix A E SE which is nearest to A^＊ in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.     

Symplectic orthogonality spaces

Let E be a finite dimensional vector space over the Galois field GF(2). Let lin(E) denote the set of one-dimensional subspaces of E. Let Φ be the symplectic inner product on E. Consider the elements of lin(E) as vertices of a graph, two vertices being connected exactly when they are distinct and orthogonal with respect to Φ. This graph is characterized abstractly.     

Vector bundles on a three-dimensional neighborhood of a ruled surface

Let S be a ruled surface inside a smooth threefold W and let E be a vector bundle on a formal neighborhood of S. We find minimal conditions under which the local moduli space of E is finite dimensional and smooth. Moreover, we show that E is a flat limit of a flat family of vector bundles whose general element we describe explicitly.     

Boundaries and complements of infinite- dimensional manifolds in the model space

Let M be a manifold modeled on a locally convex linear metric space E≌E^ω (or ≌E^ω_f and N a Z-submanifold of M. Then N is collared in M. In this paper, we study the following problem [1, 3]: Under what conditions can M be embedded in E so that N is the topological boundary of M in E? We gain a more mild sufficient condition than the previous papers [7, 8] and a necessary and sufficient condition in the case M has the homotopy type of Sⁿ (and each component of N is simply connected if n?2) and in the case N has the homotopy type of Sⁿ (n?2). Also we obtain a necessary and sufficient condition under which M can be embedded in E so that bd M = N and cl(E\M) has the homotopy type of Sⁿ (we assume that M and N are simply connected if n ? 2).     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

Nonparametric prediction of a Hilbert space valued random variable

Let $\text{ξ}{}_{\mathrm{t}}\text{, t ?}\text{R}$, be a Markovian, measurable, strictly stationary process taking values in a measurable space (E, $\text{B}$), and g a mapping from E into a separable Hilbert space H. A statistical nonparametric predictor of g(ξ_T+h) is studied in the paper. That predictor, based on the observations of the process between the times O and T generalizes the ‘predictogram’; its asymptotic consistency is proved and some applications are given.     

The existence of non-degenerate functions on a compact differentiablem-manifoldM

Summary Let M be a compact differentiable m-manifold of class C^m in E_n, n=2m+1. Let x=(x₁, ..., x_n) represent a point in E_n. The union of the direction c on the direction sphere S_n−1 in E_n such that the scalar product c · x defines a non-degenerate fonction on M is an open subset of S_n−1 whose complement θ has a Lebesgue measure zero on S_n−1. When M is non-compact θ can be everywhere dense on S_n−1, but still has Lebesgue measure zero. To Giovanni Sansone on his 70^th birth day.     

An infraconsonant nonconsonant completely regular space

Let R denote the real numbers. We construct in ZFC a countable space X such that X has exactly one non-isolated point, X is infraconsonant, and X is not consonant. We conclude that X is a completely regular space such that Isbell topology on C(X,R) is a group topology that coincides with the natural (finest splitting) topology on C(X,R), but the Isbell and compact-open topologies on C(X,R) do not coincide. The example answers two open problems in the literature.     

A minimization problem concerning subsets of a finite set

Let F be the set of subsets of a finite set S, and for H ? F, let H′ denote the elements of F which are contained in some element of H. Given integers m_l and m_l+1 does there exist a subset H of F consisting of exactly m_ll-element subsets of S and m_l+1 (l+1)-element subsets of S such that no two elements of H are related by set-wise inclusion, and if such sets H do exist what the smallest |(l?1)(H′)| can be, where |(l?1)(H′)| is the number of (l?1)-element subsets of S in H′? A generalization of this problem, which was posed by G. Katona, is solved in this paper with the help of the generalized Macaulay theorem [2].     

Divergence theorems in path space III: Hypoelliptic diffusions and beyond

Let x denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of x. In this article, this method is extended to yield similar results for degenerate diffusion processes. In particular, these results apply to non-elliptic diffusions satisfying Hörmander's condition.     

The relationship between a commutative Banach algebra and its maximal ideal space

Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B^?1 denote the invertible group in B and id B denote the set of idempotents in B. Let [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] denote the set of path components of $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$, and [Δ, B^?1] denote the set of homotopy classes of continuous maps of Δ into B^?1. We prove that the Gelfand transform on A induces a bijection of [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] onto [Δ, B^?1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of [$\text{id}\text{(A}\text{}\text{?}\text{bo B)}$] onto [Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.     

Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings

Let E be a real Banach space, K a closed convex nonempty subset of E. Let be m total asymptotically nonexpansive mappings. An iterative sequence for approximation of common fixed points (assuming existence) of T₁,T₂,…,Tm is constructed; necessary and sufficient conditions for the convergence of the scheme to a common fixed point of the mappings are given. Furthermore, in the case that E is uniformly convex, a sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is established.     

A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted m_H, and prove that the value of this invariant is connected to the order of S. In the case where char k?=?0, it is shown that if S has finite order then it is either the identity or has order 2?m_H. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if m_H is finite. We also consider the case where char k?=?p?>?0, generalizing the results of [7] to the infinite-dimensional setting.     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions

Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.