Symbolic Domain Decomposition

TL;DR

This paper introduces a method using hybrid sets to efficiently perform symbolic domain decompositions, simplifying computations and reasoning in mathematical problems involving piecewise functions and matrices.

Contribution

It presents a novel approach employing hybrid sets for symbolic domain decomposition, reducing computational complexity from exponential to linear.

Findings

Efficient symbolic domain decomposition using hybrid sets.

Reduction of complexity in operations on piecewise functions and matrices.

Simplified reasoning in symbolic mathematical problems.

Abstract

Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity.