Showing papers in "Indiana University Mathematics Journal in 1957"

A Markovian Decision Process

Abstract: : The purpose of this paper is to discuss the asymptotic behavior of the sequence (f sub n(i)) generated by a nonlinear recurrence relation. This problem arises in connection with an equipment replacement problem, cf. S. Dreyfus, A Note on an Industrial Replacement Process.

Measures on the Closed Subspaces of a Hilbert Space

Abstract: In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function μ which assigns to every closed subspace a non-negative real number such that if {Ai} is a countable collection of mutually orthogonal subspaces having closed linear span B, then $$ \mu (B) = \sum {\mu \left( {{A_i}} \right)} $$ .

On Jordan Algebras with Two Generators

Abstract: A (non-associative) algebra \( \mathfrak{A} \) is called a Jordan algebra if its multiplication satisfies the identities $$ ab = ba,\;\left( {{a^2}b} \right)a = {a^2}\left( {ba} \right) $$ (1.1) We shall assume that the base ring Φ or \( \mathfrak{A} \) is a field of characteristic not two. If \( \mathfrak{A} \) is an associative algebra over a field of characteristic not two and multiplication composition (×), then \( \mathfrak{A} \) defines a Jordan algebra \( {\mathfrak{A}^{+} } \) relative to the Jordan multiplication ab = 1/2(a × b + b × a). A Jordan algebra is called special if it is isomorphic to a subalgebra of an algebra \( {\mathfrak{A}^{+} } \), \( \mathfrak{A} \) associative. It has been known for a long time that there exist exceptional (non-special) Jordan algebras and it has been shown by P. Cohn [2] that there exist special Jordan algebras with homomorphic images which are not special.