The position value for union stable systems

In this paper, we study the position value for games in which partial cooperation exist, that is based on a union stable coalition system. The concept of basis is introduced for these systems, allowing for a definition of the position value. Moreover, an axiomatic characterization of the position value is provided for a specific class of union stable systems. Conditions under which convexity is inherited from the underlying game to the conference game, and the position value is a core vector of the restricted game are provided.     

New spaces of functions and hyperfunctions for Hankel transforms and convolutions

In this paper we study the Hankel transformation and convolution on certain spaces of entire functions and its dual that is a space of hyperfunctions and contains the (even)-Schwartz space S _e ′. We prove that the Hankel transform is an automorphism of . Also the Hankel convolutors of are investigated. Authors’ addresses: Jorge J. Betancor, Claudio Jerez and Lourdes Rodríguez-Mesa, Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271 La Laguna (Sta. Cruz de Tenerife), Espa?a; Sandra M. Molina, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350 (7600), Mar del Plata, Argentina     

When is the union of two unit intervals a self-similar set satisfying the open set condition?

Let U _λ be the union of two unit intervals with gap λ. We show that U _λ is a self-similar set satisfying the open set condition if and only if U _λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations of the set of all such λ’s, as well as the corresponding tiling patterns, are given. The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104).     

Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R ² or a cylinder R × S ¹, which generalizes a theorem due to Fischer-Colbrie and Schoen [12]. The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L ²-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21]. Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea. Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea     

Distributional Fourier Transform and Convolution Associated to Chébli-Trimèche Hypergroups

 In this paper we investigate the convolution and the generalized Fourier transform related to Chébli-Trimèche hypergroups on new spaces of distributions. Boundedness, smoothness, uniqueness, and inversion theorems are established for this transform, as well as the main properties of the convolution. The theory developed is used in solving a differential equation involving a singular differential operator. (Received 26 January 2000; in final form 24 July 2001)     

Empirical multifractal moment measures of self-similar measure for <Emphasis Type="Italic">q</Emphasis> < 0<Superscript>*</Superscript>

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L ^q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L ^q - spectrum for q < 0 is generally considered significantly more difficult since the L ^q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L ^q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0. Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China     

Topological dimensions and multifractal Rényi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension and the lower and upper q-Rényi dimensions and of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al. Author’s address: Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland     

A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.     

Mappings of Ruled Surfaces in Simply Isotropic Space <Emphasis Type="Italic">I</Emphasis><Subscript>3</Subscript><Superscript>1</Superscript> that Preserve the Generators

In this paper we study some mappings of skew ruled surfaces in simply isotropic space which preserve the generators. We study isometries, conformal mappings and mappings which preserve the area. Furthermore, we study mappings of surfaces in I ₃ ¹ which preserve the asymptotic lines.Received December 18, 2001; in revised form July 12, 2002 Published online April 4, 2003     

Behavior of the Bergman Kernel Near Smooth Convex Boundary Points

The nontangential behavior of the Bergman metric near a smooth convex boundary point of a bounded pseudoconvex domain D ⁿ is studied in terms of its multitype.Received January 25, 2002; in revised form June 20, 2002 Published online November 18, 2002     

On the false pole problem

We prove that if a convex body has an interior false pole with respect to some hyperplane, then the body is an ellipsoid. This research was partially carried out during the postdoctoral visit of this author at University College London, and it was supported by CONACYT, México.     

On the Range of the Chébli–Trimèche Transform

We first characterise the L²-Schwartz functions whose image under the Chébli–Trimèche transform are compactly supported smooth functions. We then generalise a theorem by H. H. Bang, characterising the smooth L^p-functions whose (distributional) transform have compact support.The author is supported by a research grant from the Australian Research Council.     

Intersections of Sets Starshaped Via Paths of Bounded Length

 Let S be a nonempty closed, simply connected set in the plane. For α > 0, let ℳ denote the family of all maximal subsets of S which are starshaped via paths of length at most α. Then ⋂{M : M in ℳ} is either starshaped via α-paths or empty. The result fails without the simple connectedness condition. However, even with a simple connectedness requirement, there is no Helly theorem for intersections of sets which are starshaped via α-paths. Received November 19, 2001; in revised form April 25, 2002 Published online November 18, 2002     

Stopping games with randomized strategies

We study stopping games in the setup of Neveu. We prove the existence of a uniform value (in a sense defined below), by allowing the players to use randomized strategies. In constrast with previous work, we make no comparison assumption on the payoff processes. Moreover, we prove that the value is the limit of discounted values, and we construct ε-optimal strategies. Received: 10 May 1999 / Revised version: 18 May 2000 / Published online: 15 February 2001     

Linear Weingarten Surfaces in ℝ<Superscript>3</Superscript>

 In this paper we study properties of linear Weingarten immersions and graphs related to non-existence problems and behaviour of its curvatures. The main results are obtained giving a harmonic representation of linear Weingarten surfaces and by proving optimal estimates of the height and curvatures that the immersion must satisfy, characterizing the spherical caps as the only ones achieving these bounds. Received January 25, 2001; in revised form April 4, 2002     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Scaling properties of Hausdorff and packing measures

Let . Let be a continuous increasing function defined on , for which and is a decreasing function of t. Let be a norm on , and let , , denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions such that the corresponding Hausdorff or packing measure scales with exponent by showing it must be of the form , where L is slowly varying. We also show that for continuous increasing functions and defined on , for which , is either trivially true or false: we show that if , then for a constant c, where is the Lebesgue measure on . Received June 17, 2000 / Accepted September 6, 2000 / Published online March 12, 2001