Dividerat

Dividerat is a theoretical construct in computer science and mathematics used to describe a method for dividing a sequence into equal-width blocks according to a fixed divisor. For a positive integer d, the dividerat function divat(n, d) yields a pair (block, offset), where block = floor(n / d) and offset = n mod d. The pair uniquely identifies the position of the element with index n within blocks of size d.

In essence, dividerat is a formal name for the quotient-remainder representation of integer division, used here to emphasize the block-based view of indexing. The concept helps describe how a sequence can be partitioned into consecutive blocks, enabling a consistent way to refer to the position of any element by its block and its offset within that block.

Practical use of the dividerat mapping includes data partitioning for parallel processing, organizing memory layouts to improve locality, and constructing cache-friendly data structures. For example, with d = 3, indices 0, 1, and 2 fall in block 0 with offsets 0, 1, and 2, while indices 3, 4, and 5 fall in block 1 with offsets 0, 1, and 2.

Notes and context: dividerat is not a standard operator in mainstream mathematics; it is a descriptive term used in theoretical discussions or exploratory work to illustrate block decomposition and scheduling concepts. Related ideas include division algorithms, modular arithmetic, array partitioning, and load balancing.