Rado equations solved by linear combinations of idempotent ultrafilters

Idempotent Ultrafilters and Polynomial Recurrence

In the thirty or so years since H. Furstenberg reproved Szemerédi’s theorem using methods from ergodic theory, many striking discoveries have been made in the area now known as Ergodic Ramsey theory. Perhaps the most surprising of these is the discovery that recurrence results can be obtained for polynomial sets, meaning sets of values of polynomials. The following pretty theorem, a special cas...

Linear Ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are ‘‘continuous’’ with respect to U modulo the ideal of those that are ‘‘trivial’’ with respect to U forms a division ring E(U ). (These division rings can also be described as the endomorphism rings of the simple left End( X )-modules.) We study this and the dual construction, ...

Solution of Generalized Linear Vector Equations in Idempotent Algebra

The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of equations are obtained. The results are exte...

Solution of linear equations and inequalities in idempotent vector spaces

Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniquen...

Linear Analogues of Ultrafilters