This work describes the proof and uses of a theorem allowing definition of recursive functions over the type of λ-calculus terms, where terms with bound variables are identified up to α-equivalence. The theorem embodies what is effectively a principle of primitive recursion, and the analogues of this theorem for other types with binders are clear. The theorem’s side-conditions require that the ...

Many of us who teach calculus and mathematical topics that use calculus have taken for granted that hand-held calculators use Taylor series or a variant to compute transcendental functions. Thus, it was a surprise to learn that this was not the case. The CORDIC method (Coordinate Rotation Digital Computer) was developed by Jack Volder [6] in the late 1950’s. Hewlett-Packard was quick to realize...

The goal of this paper is to introduce the RCFA (Recursive Cost Function Adaptation) algorithm. The derivation of the new algorithm does not use an estimator of the instantaneous error as the previous CFA (Cost Function Adaptation) algorithms did. In the RCFA case, the new error power is computing from the previous error power using an usual LMS recursive equation. The proposed method improves ...

We prove in this paper an effective recursion formula for all intersection numbers of mixed ψ and κ classes on moduli spaces of curves. Several related general recursive formulas are also proved. Some of these formulas contain certain tautological constants, which are given by the secant Euler numbers and series expansion of Bessel functions of the first kind.

In this paper, I show that general recursive definitions can be represented in the free monad which supports the ‘effect’ of making a recursive call, without saying how these calls should be executed. Diverse semantics can be given within a total framework by suitable monad morphisms. The Bove-Capretta construction of the domain of a general recursive function can be presented datatype-generica...

In this paper we discuss some recent results on two diierent types of growth of Lie algebras, that lead to some combinatorial problems. First, we study the growth of nitely generated Lie algebras (sections 1{4). This problem leads to a study of generalized partitions. Recently the author has suggested a series of q-dimensions of algebras Dim q , q 2 N which includes, as rst terms, dimensions of...

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of i...

The work in this paper is inspired and motivated by some work of Nathanson. We count the number of relatively prime subsets and the number of relatively prime subsets having some fixed cardinality that are in {m,m+1, . . . , n}. We also count the number of nonempty subsets of {m,m+1, . . . , n} whose gcd is relatively prime to n and the number of nonempty subsets {m,m+1, . . . , n} having some ...

We consider a multidimensional diffusion X with drift coefficient b(Xt, α) and diffusion coefficient εa(Xt, β) where α and β are two unknown parameters, while ε is known. For a high-frequency sample of observations of the diffusion at the time points k/n, k = 1, . . . , n, we propose a class of contrast functions and thus obtain estimators of (α, β). The estimators are shown to be consistent an...

A fundamental requirement of any vision system designed to perform recognition tasks is a library of representations of objects it is likely to encounter. The classic approach to constructing such representations is to decompose the shape into simple primitive elements. However, for complex shapes it is usually necessary to use a large number of such primitives to achieve an accurate decomposit...