Kakutani

In mathematics, "Kakutani" may refer to:

• Shizuo Kakutani (1911-2004): A Japanese-American mathematician best known for his contributions to functional analysis, operator theory, ergodic theory, and probability theory. Much of the following is named in his honor.

• Kakutani's Fixed Point Theorem: A fixed-point theorem that guarantees the existence of a fixed point for set-valued functions satisfying certain conditions. Specifically, if S is a nonempty, compact, convex subset of a Euclidean space, and f: S → 2^S is an upper hemicontinuous set-valued function with the property that f(x) is nonempty, closed, and convex for all x in S, then f has a fixed point (i.e., there exists an x in S such that x is an element of f(x)). This theorem is a generalization of Brouwer's fixed-point theorem and is widely used in mathematical economics and game theory to prove the existence of Nash equilibria.

• Kakutani's Problem (also known as the 3n+1 problem, or the Collatz conjecture): Although Shizuo Kakutani is often mistakenly attributed to originating the 3n+1 problem, there is no evidence that he was the originator or made significant contributions to it. The 3n+1 problem postulates that for any positive integer n, the sequence defined by n → n/2 if n is even, and n → 3n+1 if n is odd, will always eventually reach 1. It remains an unsolved problem in mathematics.

• Kakutani Tower: In ergodic theory, a Kakutani tower (or Rokhlin tower) is a construct used to visualize and analyze measure-preserving transformations. It consists of a base set and a series of "towers" built upon it by repeatedly applying the transformation. It is a key tool in understanding the ergodic properties of dynamical systems.