We derive a formula for an n-th order divided difference of the inverse of a function. The formula has a simple and surprising structure: it is a sum over partitions of a convex polygon with n+1 vertices. The formula provides a numerically stable method of computing divided differences of k-th roots. It also provides a new way of enumerating all partitions of a convex polygon of a certain type,...

This study examined the effect of divided attention (DA) on global judgment of learning (JOL) accuracy in a multitrial list learning paradigm. A word monitoring task was used to divide attention. Participants were assigned to an attention condition (DA at encoding, DA at judgment, DA at retrieval, or focused attention) and completed 4 learning trials, each comprising a study, judgment, and reca...

Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theore...

In this paper we obtain a formula which involves the n th derivative of the first order divided difference and a corresponding inequality for functions whose (n + 1)-th derivative belongs to a L p space. These results are a generalization of the results from [1] and [2]. Finally, some examples are given.

holds, with E(f, x) explicitly given as an integral involving Df . Since, for his particular choice of L, the sum involves the derivatives of f at the points or nodes xi associated with L, Han thinks of this as a ‘multinode’ expansion for f . It is the purpose of this note to point out that this asymptotic error expansion, properly interpreted, holds for any bounded linear map L on C([a . . b])...

Abstmet. A method of surface definition and smoothing using a set of three-dimensional skeletal-lines is given. The skeletal-lines represent the essential features of a surface and can be individually smoothed using the first divided difference. The modified skeletal-lines are automatically generated by back substitution from the smoothed first divided difference lines. In this way a surface ca...

The notion of two dimensional divided difference was introduced by T. Popoviciu in 1934. Many properties of these differences were obtained by D. V. Ionescu. Other properties of the mentioned differences were obtained by the authors of the present paper. The focus of the present paper is to establish properties of two dimensional differences in the case of multiple knots. First, we establish so...

This paper is devoted to the semilocal convergence, using centered hypotheses, of a third order Newton-type method in a Banach space setting. The method is free of bilinear operators and then interesting for the solution of systems of equations. Without imposing any type of Fréchet differentiability on the operator, a variant using divided differences is also analyzed. A variant of the method u...

We show that every regular language, via some DFA which accepts it, gives rise to a homomorphism from the semiring of polynomial functors to the semiring of n×n matrices over polynomial functors. Given some polynomial functor and a regular language, this homomorphism can be used to automatically derive a functor whose values have the same shape as those of the original functor, but whose sequen...

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t1, t2, . . . , tn]. We show these group actions are the same as an action studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and ot...