We propose a progressive approach to simplify teaching recursion. It is introduced by means of three instances of increasing level of difficulty. First, recursion is introduced in a “pure” form as it is found in formal grammars; its accompanying mechanisms are related to rewriting. Later, a functional programming language allows defining recursive functions and recursive data types; its mechani...

This paper provides an overview of the formal semantics of VDM-SL which currently is being standardised by ISO. This is a specification language used in the formal method known as the Vienna Development Method (or simply VDM). In this paper we will focus on the foundations and the semantics of a rather unique combination of looseness and recursion.

In this article we review the Adomian decomposition method (ADM) and its modifications including different modified and parametrized recursion schemes, the multistage ADM for initial value problems as well as the multistage ADM for boundary value problems, new developments of the method and its applications to linear or nonlinear and ordinary or partial differential equations, including fractio...

Everybody knows what recursion is and how it should be implemented in a procedural language. The first analysis of recursion implementations is due Kent Beck [Beck92], but no formalization was given. The other work on an OOP implementation of the subject is due to Boby Wolf [Wolf97], but his work only deals with one aspect of the problem: Structural Recursion. Structural Recursion is a particul...

We propose a method for encoding iterators (and recursion operators in general) using interaction nets (INs). There are two main applications for this: the method can be used to obtain a visual notation for functional programs; and it can be used to extend the existing translations of the λ-calculus into INs to languages with recursive types.

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice “inversion relation” method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site Tpq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of Tpq. In terms...

We apply a tree-based methodology to solve new, very broadly defined families of nested recursions of the general form R(n) = ∑k i=1R(n − ai − ∑p j=1R(n − bij)), where ai are integers, bij are natural numbers, and k, p are natural numbers that we use to denote “arity” and “order,” respectively, and with some specified initial conditions. The key idea of the tree-based solution method is to asso...

Dyson’s hierarchical model (HM) is a lattice scalar model for which the effective potential can be calculated very accurately using the renormalization group method. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigorous and numerical result...

The perfect quenching of spin tunneling that has previously been discussed in terms of interfering instantons, and has recently been observed in the magnetic molecule Fe8, is treated using a discrete phase integral (or Wentzel-KramersBrillouin) method. The simplest model Hamiltonian for the phenomenon leads to a Schrödinger equation that is a five-term recursion relation. This recursion relatio...

In these notes we look at the related techniques of recursion and induction. You will have seen already some of the simpler kinds of induction (over N) and you may have met some forms of recursion (such as primitive recursion) but you may not be aware of this name. Here we will go beyond these rather simple particular cases, and look at some more intricate aspects. Roughly speaking (at least fo...