Interpolation on lattices generated by cubic pencils

Principal lattices are distributions of points in the plane obtained from a triangle by drawing equidistant parallel lines to the sides and taking the intersection points as nodes. Interpolation on principal lattices leads to particularly simple formulae. These sets were generalized by Lee and Phillips considering three-pencil lattices, generated by three linear pencils. Inspired by the addition of points on cubic curves and using duality, we introduce an addition of lines as a way of constructing lattices generated by cubic pencils. They include three-pencil lattices and then principal lattices. Interpolation on lattices generated by cubic pencils has the same good properties and simple formulae as on principal lattices. Dedicated to C.A. Micchelli for his mathematical contributions and friendship on occasion of his sixtieth birthday Mathematics subject classifications (2000) 41A05, 41A63, 65D05. J.M. Carnicer: Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Error estimates for interpolation of rough data using the scattered shifts of a radial basis function

The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function – the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough. AMS subject classification 41A05, 41A25, 41A30, 41A63R.A. Brownlee: Supported by a studentship from the Engineering and Physical Sciences Research Council.     

A structural characterization of non-Arguesian lattices

This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by badly gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a classical non-Arguesian configuration.Research supported by NSERC Operating Grant A8190.Research supported by NSF Grant DMS-8300107.     

Maeda-type Decomposition of CJ-generated Algebraic Lattices

We prove necessary and sufficient conditions for the decomposition of an arbitrary CJ-generated algebraic lattice into a direct product of subdirectly irreducible lattices. We generalize earlier results due to F. Maeda, T. Katriák and the present author. New structure theorems for two classes of CJ-generated algebraic lattices are also obtained.AMS Subject Classification (1991) 06B05 06B10Research partially supported by Hungarian National foundation for Scientific Research, Grant No. T029525 and T030243 and by János Bolyai Grant of Hungarian Academy of Science.     

Modelling of Ring Geometry from von Neumann’s Point of View

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.     

Near-optimal data-independent point locations for radial basis function interpolation

The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples. AMS subject classification 41A05, 41063, 41065, 65D05, 65D15This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen.     

Generalized principal lattices and cubic pencils

Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4. Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.     

Lattice structures and spreading models

We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semi-lattices and countable lattices and all finite lattices. Research of the second named author was partially supported by the National Science Foundation. The third named author had a visiting appointment at the University of South Carolina for the 2004–05 academic year during part of his research.     

Dilworth's principal elements

Principal elements were introduced in multiplicative lattices by R. P. Dilworth, following an earlier but less successful attempt in the joint work of Ward and Dilworth. As suggested by their name, principal elements are the analogue in multiplicative lattices of principal ideals in (commutative) rings. Principal elements are the cornerstone on which the theory of multiplicative lattices and abstract ideal theory now largely rests. In this paper, we obtain some new results regarding principal elements and extend some others. In addition, we try to convey what is known and what is not known about the subject. We conclude with a fairly extensive (but likely not exhaustive) bibliography on principal elements.Dedicated to R. P. DilworthPresented by E. T. Schmidt.     

Cubic pencils of lines and bivariate interpolation

Cubic pencils of lines are classified up to projectivities. Explicit formulae for the addition of lines on the set of nonsingular lines of the pencils are given. These formulae can be used for constructing planar generalized principal lattices, which are sets of points giving rise to simple Lagrange formulae in bivariate interpolation. Special attention is paid to the irreducible nonsingular case, where elliptic functions are used in order to express the addition in a natural form.     

Ultraconvergence of ZZ patch recovery at mesh symmetry points

Summary. The ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least-squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed and numerical examples are demonstrated. Mathematics Subject Classification (2000):65N30, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25This research was partially supported by the National Science Foundation grants DMS-0074301, DMS-0079743, and INT-0196139     

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Padé approximations to the logarithm III: Alternative methods and additional results

We use C. Schneider’s summation software Sigma and a method due to Andrews-Newton-Zeilberger to reprove results from [5] and [24] as well as to prove new related material. 2000 Mathematics Subject Classification Primary—41A21, 05A19, 33F10 K. Driver: Supported by NRF-Grant 2047226. H. Prodinger: Supported by NRF-Grant 2053748. C. Schneider: Supported by the Austrian Academy of Sciences, by the John Knopfmacher Research Centre for Applicable Analysis and Number Theory, and by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF. J. A. C. Weideman: Supported by NRF-Grant FA2005032300018.     

Projective spaces on partially ordered sets and Desargues' postulate

We introduce a generalized concept of projective and Desarguean space where points (and lines) may be of different size. Every unitary module yields an example when we take the 1-and 2-generated submodules as points and lines. In this paper we develop a method of constructing a wide range of projective and Desarguean spaces by means of lattices.     

Boolean products of lattices

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.Research supported by the Natural Sciences and Engineering Research Council of Canada.Research supported by ONR Grant N00014-90-J-1008.     

Loops mit distributiven und geometrischen Unterloopverbänden

The following facts are shown: A loop with a finite distributive subloop lattice is finite, monogenic and all its subloops are monogenic. Therefore, power-associative loops having finite distributive subloop lattices are cyclic groups. A loop G with its subloop lattice L(G) being a finite n-dimensional projective geometry is generated by at most n+1 elements. For all n IN, n4, there are power-associative loops whose subloop lattices are projective lines with n points. Furthermore, for a given projective planeP _n (desarguesian or non-desarguesian) of order n there exists a power-associative loopG with L(G) -P _n.     

On Linear Combinatorics III. Few Directions and Distorted Lattices

Our main tools will be certain ``distorted' lattices which contain many rich lines (i.e. straight lines incident upon many points of the lattice). Received: November 28, 1996     

Some remarks concerning the infinite distributivity of lattices

Theorems are proved concerning the regular imbedding of an infinitely distributive lattice in least decomposable lattices and in completely decomposable lattices. A relation is established between decomposable lattices, Stone lattices, and completely decomposable lattices.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 95–103, July, 1970.We use this opportunity to thank A. G. Pinsker for his valuable advice concerning this work.     

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation

Combinatorial identities that were needed in [25] are proved, mostly with C. Schneider’s computer algebra package Sigma. The form of the Padé approximation of the logarithm of arbitrary order is stated as a conjecture. 2000 Mathematics Subject Classification Primary—41A21, 05A19, 33F10 Supported by NRF-grant 2047226. Supported by NRF-grant 2053748. Supported by the Austrian Academy of Sciences, by the John Knopfmacher Research Centre for Applicable Analysis and Number Theory, and by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF. Supported by NRF-grant 2053756.