intervalares

Intervalares are a mathematical construct used to represent uncertain quantities by collecting multiple plausible intervals into a single object. An intervalare is a finite collection of closed real intervals, denoted I = { [a1, b1], [a2, b2], ..., [an, bn] }. It expresses that a quantity may lie in any of its constituent intervals, capturing ambiguity, multiple scenarios, or layered information about dependency among measurements. To handle dependencies, an intervalare can be paired with a dependency structure that links intervals originating from the same source, helping to avoid overly conservative results in subsequent calculations.

Operations on intervalares extend standard interval arithmetic in a componentwise fashion. The sum A + B of intervalares A = { [ai, bi] } and B = { [cj, dj] } is the set { [ai + cj, bi + dj] : i = 1..n, j = 1..m }, followed by a reduction step that merges overlapping or adjacent intervals and removes empty results. Similar componentwise rules apply for subtraction, multiplication, and division, with appropriate handling of edge cases such as division by intervals containing zero. Union and intersection are defined by applying the respective interval operations to all combinations of components and then reducing the result.

Applications of intervalares include robust optimization, uncertainty propagation in numerical simulations, constraint solving under ambiguity, and sensor fusion where measurements yield multiple plausible ranges. They generalize single-interval representations and provide a framework for tracking alternative scenarios and dependencies within a unified structure. See also interval arithmetic, interval graphs, and affine arithmetic.