Maximal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>‐regularity for elliptic operators with VMO‐coefficients

In this paper we show that the parabolic problem associated with elliptic operators A having coefficients in has the property of maximal -regularity. RID="h1" ID="h1"In Memoriam Philippe Bénilan     

On spectral geometry of minimal parallel submanifolds

We consider the problem of characterizing some minimal submanifolds using the spectrum⁰Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCP ⁿ, 3≤n≤6, by their⁰Spec in the class of all compact totally real minimal submanifolds ofCP ⁿ. Moreover, we characterize the Clifford torus by its⁰Spec in the class of all compact minimal submanifolds of the Euclidean sphereS ⁿ⁺¹(1). Authors supported by funds of the University of Lecce and the M.U.R.S.T.     

Some Periodic and Non-Periodic Recursions

 It is well known that the recurrence relations

Another Look at the Erdős–Hajnal–Pósa Results on Partitioning Edges of the Rado Graph

Dedicated to the memory of Paul Erdős Erdős, Hajnal and Pósa exhibited in [1] a partition (U,D) of the edges of the Rado graph which is a counterexample to . They also obtained that if every vertex of a graph has either in or in the complement of finite degree then . We will characterize all graphs so that . Received October 29, 1999 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.     

Density Conditions for Panchromatic Colourings of Hypergraphs

Let be a hypergraph. A panchromatic t-colouring of is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of is . Among other results, it is proved here that if every edge has at least t vertices and whenever , then is panchromatically t-choosable, and this condition is sharp; the minimum such that every t-uniform hypergraph with is panchromatically t-choosable satisfies ; and except possibly when t = 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if whenever , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree. Received November 10, 1998 RID="*" ID="*" This work was carried out while the first author was visiting Nottingham, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.     

Existence of Polyhedral Embeddings of Graphs

It is proved that the decision problem about the existence of an embedding of face-width 3 of a given graph is NP-complete. A similar result is proved for some related decision problems. This solves a problem raised by Neil Robertson. Received July 6, 1998 RID="*" ID="*" Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1–0502–0101–98.     

The Dimension of Subsets of Moran Sets Determined by the Success Run Behaviour of Their Codings

 Let be a Moran set associated with the set . Let Γ be a non-empty subset of with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space is a decomposition of F such that

Haar Function Based Estimates of the Star-Discrepancy of Plane Digital Nets

 We apply the Haar function system to estimate the star-discrepancy of special digital (t,m,s)-nets in dimension . We use a basic technique based on discretization combined with an exact calculation of the discrete star-discrepancy. (Received 30 August 1999; in revised form 17 January 2000)     

On the correlation of pseudorandom numbers generated by inversive methods

We introduce two new types of inversive generators for pseudorandom numbers. These new methods offer several advantages over the conventional inversive generator. For instance, we establish good correlation properties of our generators that cannot be obtained for the conventional inversive generator with current methods. A new bound on character sums for finite fields is the essential technical tool for this work. Authors’ addresses: Harald Niederreiter, Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore; Jo?l Rivat, Institut de Mathématiques de Luminy, Université de la Méditerranée, CNRS-UMR 6206, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France     

A method for the resolution of the Jacobi equation <Emphasis Type="Italic">Y</Emphasis>″ + <Emphasis Type="Italic">RY</Emphasis> = 0 on the manifold <Emphasis Type="Italic">Sp</Emphasis>(2)/<Emphasis Type="Italic">SU</Emphasis>(2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls. Work partially supported by DGI (Spain) and FEDER Projects MTM 2004-06015-C02-01 and MTM 2007-65852 (first author) and by Research Project PGIDIT05PXIB16601PR (second author). Authors’ addresses: A. M. Naveira, Departamento de Geometría y Topología. Facultad de Matemáticas, Avda. Andrés Estellés, N1, 46100 – Burjassot, Valencia, Spain; A. D. Tarrío Tobar, E. U. Arquitectura Técnica, Campus A Zapateira. Universidad de A Coru?a, 15192 – A Coru?a, Spain