The general Neville-Aitken-algorithm and some applications

Summary In this note we will present the most general linear form of a Neville-Aitken-algorithm for interpolation of functions by linear combinations of functions forming a ebyev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkena-algorithm[8]applying to complete ebyev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given.     

Remarks on a unified theory for classical and generalized interpolation and extrapolation

A unified theory for generalized interpolation, as developed by Mühlbach, and classical polynomial interpolation is discussed. A fundamental theorem for generalized linear iterative interpolation is given and used to derive generalizations of the classical formulae due to Neville, Aitken and Lagrange. Using Mühlbach's definition of generalized divided differences, Newton's generalized interpolation formula, including an expression for the error term, is derived as a pure identity.     

Generalized outer synchronization between non-dissipatively coupled complex networks with different-dimensional nodes

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.     

The Generalized Moment Problem with Complexity Constraint

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.     

Quaternion orthogonal designs from complex companion designs

The success of applying generalized complex orthogonal designs as space-time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space-time-polarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build quaternion orthogonal designs on complex variables for any even number of columns. A second related technique is applied to maximum rate complex orthogonal designs to generate an infinite family of quaternion orthogonal designs on complex variables such that the resulting designs have no zero entries. This second technique is also used to generate an infinite family of quaternion orthogonal designs defined over quaternion variables that display a regular redundancy. The proposed constructions are theoretically important because they provide the first known direct techniques for building infinite families of orthogonal designs over quaternion variables for any number of columns.     

Generalized harmonic numbers with Riordan arrays

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.     

A maximum principle for node voltages in finitely structured,transfinite, electrical networks

Transfinite electrical networks have unique finite-powered voltage-current regimes given in terms of branch voltages and branch currents, but they do not in general possess unique node voltages. However, if their structures are sufficiently restricted, those node voltages will exist and will satisfy a maximum principle much like that which holds for ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalized the idea of local-finiteness. Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's current laws for nodes of any ranks and the permissibility of connecting pure voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a subsequent work.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

Hyperfinite stochastic integration for Lévy processes with finite-variation jump part

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part.As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.     

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.     

Approximation by Discrete GB-Splines

This paper addresses the definition and the study of discrete generalized splines. Discrete generalized splines are continuous piecewise defined functions which meet some smoothness conditions for the first and second divided differences at the knots. They provide a generalization both of smooth generalized splines and of the classical discrete cubic splines. Completely general configurations for steps in divided differences are considered. Direct algorithms are proposed for constructing discrete generalized splines and discrete generalized B-splines (discrete GB-splines for short). Explicit formulae and recurrence relations are obtained for discrete GB-splines. Properties of discrete GB-splines and their series are studied. It is shown that discrete GB-splines form weak Chebyshev systems and that series of discrete GB-splines have a variation diminishing property.     

Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

A generalized characteristic polynomial of a graph having a semifree action

For an abelian group Γ, a formula to compute the characteristic polynomial of a Γ-graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35-46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a Γ-graph when Γ is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.     

Implementation of chebyshevian linear multistep formulas

A Chebyshevian linear multistep formula is a formula fitted to a Chebyshev set of basis functions. This paper presents a unified approach for the implementation of Chebyshevian backward differentiation and Adams formulas for solving ordinary differential equations. The approach is based on generalized scaled differences, derived from generalized divided differences, and it includes the generalized Newton interpolation formula as predictor for the Chebyshevian implicit backward differentiation formula and Chebyshevian Adams-Bashforth-Moulton formulas. The local truncation errors are estimated by means of the scaled differences providing information for the control of order and steplength.     

A Newton basis for Kernel spaces

It is well known that representations of kernel-based approximants in terms of the standard basis of translated kernels are notoriously unstable. To come up with a more useful basis, we adopt the strategy known from Newton’s interpolation formula, using generalized divided differences and a recursively computable set of basis functions vanishing at increasingly many data points. The resulting basis turns out to be orthogonal in the Hilbert space in which the kernel is reproducing, and under certain assumptions it is complete and allows convergent expansions of functions into series of interpolants. Some numerical examples show that the Newton basis is much more stable than the standard basis of kernel translates.     

General recurrence and ladder relations of hypergeometric-type functions

A method for the explicit construction of general linear sum rules involving hypergeometric-type functions and their derivatives of any order is developed. This method only requires the knowledge of the coefficients of the differential equation that they satisfy, namely the hypergeometric-type differential equation. Special attention is paid to the differential-recurrence or ladder relations and to the fundamental three-term recurrence formulas. Most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials, are instances of these sum rules. Moreover, an extension of the method to the generalized hypergeometric-type functions is also described, allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen-like atoms, where the varying parameter is the dimensionality.     

Descents, inversions, and major indices in permutation groups

A multivariate generating function involving the descent, major index, and inversion statistic first given by Ira Gessel is generalized to other permutation groups. We provide generating functions for variants of these three statistics for the Weyl groups of type B and D, wreath product groups, and multiples of permutations. All of our ideas are combinatorial in nature and exploit fundamental relationships between the elementary and homogeneous symmetric functions.     

ECT-B-splines defined by generalized divided differences

ECT-spline curves are generated from different local ECT-systems via connection matrices. If they are nonsingular, lower triangular and totally positive there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized either to form a nonnegative partition of unity or to have integral one. In this paper such ECT-B-splines are defined by generalized divided differences. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines [Mühlbach and Tang, Calculation of ECT-B-splines and of ECT-spline curves recursively, in preparation].There is an ECT-spline space naturally adjoint to every ECT-spline space. We also construct B-splines via generalized divided differences for this space and study relations between the two adjoint spaces.     

A parallel method for fast and practical high-order newton interpolation

We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n ²), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.Supported in part by the National Science Foundation under Grant No. NSF DCR-8603722.Supported by the National Science Foundation under Grants No. US NSF MIP-8410110, US NSF DCR85-09970, and US NSF CCR-8717942 and AT&T under Grant AT&T AFFL67Sameh.