Array based data parallel programming can be generalized in two ways to make it an appropriate paradigm for parallel processing of general recursively de ned data. The rst is the introduction of a parallel evaluation mechanism for dynamically allocated recursively dened data. It achieves the e ect of applying the same function to all the subterms of a given datum in parallel. The second is a ne...

Depth-first search of a graph is formalized with function. It is shown that it visits all of the reachable nodes from a given list of nodes. Executable ML code of depth-first search is obtained with code generation feature of Isabelle/HOL. The formalization contains two implementations of depth-first search: one by stack and one by nested recursion. They are shown to be equivalent. The terminat...

In an earlier communication we have developed a recursion based approach to the study of phase stability and transition of binary alloys [1]. We had combined the recursion method introduced by Haydock, Heine and Kelly[2] and the our augmented space approach[3] with the orbital peeling technique proposed by Burke [4] to determine the small energy differences involved in the discussion of phase s...

This paper introduces a new notion of quantum recursion of which the control flow of the computation is quantum rather than classical as in the notions of recursion considered in the previous studies of quantum programming. A typical example is recursive quantum walks, which are obtained by slightly modifying the construction of the ordinary quantum walks. The operational and denotational seman...

Abstract Recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, corecursion underappreciated still seen as exotic. We aim to put them both on equal footing by giving foundation for primitive based computation, terminating calculus analogous original computational recursion. show how implementation details an abstract machine strengthens their conne...

Abstract. Let dμ be a nonnegative measure with support on the real axis and let α ∈ R be outside the convex hull of the support. This paper describes a new approach to determining recursion coefficients for Gauss quadrature rules associated with measures of the form dμ̌(x) := dμ(x)/(x − α)2l. The proposed method is based on determining recursion coefficients for a suitable family of orthonormal ...

This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schrödinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schrödinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed...

The recursion relations are derived for multi-photon processes of noncommutative QED. The relations concern purely photonic processes as well as the processes with two fermions involved, both for arbitrary number of photons at tree level. It is shown that despite of the dependence of noncommutative vertices on momentum, in contrast to momentum-independent color factors of QCD, the recursion rel...

We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for N = 2 and N = 3 containing the equations of shallow water waves and its generalizations with their first two higher symmetries and their recursion operators. We also discuss a reduction of N + 1 systems to N systems of so...