The Denjoy extension of the Riemann and Mcshane integrals

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval [a, b] into a Banach space X. It is shown that a Denjoy-Bochner integrable function on [a, b] is Denjoy-Riemann integrable on [a, b], that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b] and that a Denjoy-McShane integrable function on [a, b] is Denjoy-Pettis integrable on [a, b]. In addition, it is shown that for spaces that do not contain a copy of c ₀, a measurable Denjoy-McShane integrable function on [a, b] is McShane integrable on some subinterval of [a, b]. Some examples of functions that are integrable in one sense but not another are included.     

The Chern character of a transversally elliptic symbol and the equivariant index

Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G-invariant generalized function on G, constructed in terms of equivariant closed differential forms on the cotangent bundle T ^* M. Oblatum 24-VII-1995     

Inner Derivations on σ-C*-Algebras

We prove the following two improvements of a result of Becker. (1) If A is a pro-C*-algebra, then every derivation on A is approximately inner. (2) If A is a separable σ-C*-algebra, and if every C* quotient of A has the property that every derivation on it is inner, then also every derivation on A is inner. We also give an example of a derivation on a separable σ-C*-algebra which is not inner but which induces an inner derivation on every C* quotient.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

The automorphism group of a Banach principal bundle with {1}-structure

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach–Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach–Lie group acting smoothly on M.     

Quiver Hopf algebras

In this paper we study subHopfalgebras of graded Hopf algebra structures on a path coalgebra kQ^c. We prove that a Hopf structure on some subHopfquivers can be lifted to a Hopf structure on the whole Hopf quiver. If Q is a Schurian Hopf quiver, then we classified all simple-pointed subHopfalgebras of a graded Hopf structure on kQ^c. We also prove a dual Gabriel theorem for pointed Hopf algebras.     

Behavior of zeros of polynomials of near best approximation

The purpose of this paper is to study the asymptotic behavior of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example, suppose E is a finite union of compact intervals of the real line and f is a continuous function on E, but is not analytic on E; then we show (cf. Corollary 2.2) that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. This fact answers a question posed by P. Borwein who showed that, for the case when E is a single interval and f is real-valued, then the above hypotheses on f imply that at least one point of E is the limit point of zeros of such polynomials.     

Invariant Metrics on G-Spaces

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Invariant measures and disintegrations with applications to Palm and related kernels

Consider a locally compact group G acting measurably on some spaces S and T. We prove a general representation of G-invariant measures on S and the existence of invariant disintegrations of jointly invariant measures on S × T. The results are applied to Palm and related kernels associated with a stationary random pair (ξ,η), where ξ is a random measure on S and η is a random element in T. An erratum to this article can be found at     

Towards Bounding Complexity of a Minimal Model

We give some effectivity results in birational geometry. We provide an upper bound on the rational constant in Rationality Theorem in terms of certain intersection numbers, under an additional condition on the variety that it admits a divisorial contraction. One consequence is an explicit bound on the number of certain extremal rays. Our main result tries to construct from a given set of ample divisors H _j on X with their intersection numbers b _i , a certain set of ample divisors L _j on X' or X ⁺ where X' or X ⁺ arises from a contraction or a flip,such that the corresponding intersection numbers of L _j are uniformly bounded in terms of b _i and the index of X. This gives a bound on the projective degree of a minimal model in special case.     

In response to the article by Ivan Kausz: A discriminant and an upper bound for w 2 for hyperelliptic arithmetic surfaces

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.     

Maximum pebbling number of graphs of diameter three

Given a configuration of pebbles on the vertices of a graph G, a pebbling move consists of taking two pebbles off some vertex v and putting one of them back on a vertex adjacent to v. A graph is called pebbleable if for each vertex v there is a sequence of pebbling moves that would place at least one pebble on v. The pebbling number of a graph G is the smallest integer m such that G is pebbleable for every configuration of m pebbles on G. We prove that the pebbling number of a graph of diameter 3 on n vertices is no more than (3/2)n + O(1), and, by explicit construction, that the bound is sharp. © 2006 Wiley Periodicals, Inc. J Graph Theory