In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and in particular the necklace splitting problem

Necklace splitting problem

In mathematics, and in particular combinatorics, the necklace splitting problem arises in a variety of contexts including exact division; its picturesque name is due to mathematicians Noga Alon and Douglas B. West....

, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice

John R. Rice (professor)

John Richard Rice is an American mathematician and computer scientist, the W. Brooks Fortune Distinguished Professor Emeritus of Computer Science and a professor of mathematics at Purdue University...

; a simplified proof was given in 1976 by A. Pinkus.

The theorem

If

are given continuously integrable function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s then there exist

and

such that

The theorem was used by Noga Alon

Noga Alon

Noga Alon is an Israeli mathematician noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers.- Academic background :...

in the context of necklace splitting in 1987.