Carl de Boor 8oct02 Abstract The Leibniz formula, for the divided difference of a product, and Opitz’s formula, for the divided difference table of a function as the result of evaluating that function at a certain matrix, are shown to be special cases of a formula available for the coefficients, with respect to any basis, of an ‘ideal’ or ‘Hermite’ polynomial interpolant, in any number of varia...

Divided differences can be viewed as a discrete analogue of derivatives and are commonly used in approximation theory, see [1] for a survey. Recently, the second author and Lyche established two univariate chain rules for divided differences [2], both of which can be viewed as analogous to Faà di Bruno’s formula for differentiating composite functions [6,7]. One of these formulas was simultaneo...

Let V be an even dimensional vector space over a field K of characteristic 2 equipped with a non-degenerate alternating bilinear form f . The divided power algebra DV is considered as a complex with differential defined from f . We examine the cohomology modules as representations of the corresponding symplectic group.

In this paper we derive two formulas for divided differences of a function of a function. Both formulas lead to other divided difference formulas, such as reciprocal and quotient rules. The two formulas can also be used to derive Faà di Bruno’s formula and other formulas for higher derivatives of composite functions. We also derive a divided difference version of Faà di Bruno’s determinant form...

We examined how encoding and retrieval processes were affected by manipulations of attention, and whether the degree of semantic relatedness between words in the memory and distracting task modulated these effects. We also considered age and bilingual status as mediating factors. Monolingual and bilingual younger and older adults studied a list of words from a single semantic category presented...

THE CAUCHY DOUBLE ALTERNANT AND DIVIDED DIFFERENCES WENCHANG CHU Dedicated to my friend Pier Vittorio Ceccherini for his 65th birthday Abstract. As an extension of Cauchy’s double alternant, a general determinant evaluation formula is established. Several interesting determinant identities are derived as consequences by means of divided differences.

A multivariate kth divided diierence is proposed and shown to lead to new error formull for polynomial interpolation on some standard multivariate point sets. x1. Introduction In trying to understand what might be a reasonable way to describe the error in interpolation by polynomials on l R d , I came upon the following considerations: The standard error formula g(x) = g(a) + Z 1 0 D x?a g(a + ...

Two models have been advanced to account for the apparent ease with which attention can be divided in music: a `̀ divided attention’ ’ model postulates that listeners effectively manage to follow two or more melodic lines played simultaneously. According to a `̀ ® gure-ground model,’’ the harmonic coherence of Western polyphonies allows a listener to focus on one melody while staying aware of the...