An introduction to pseudo-linear algebra

Weighted automata with discounting

“…Discounting has been intensively investigated in Markov decision processes, economics, and game theory [FV97,Sha53]. Related multiplications of series have also been considered in the area of Ore series in difference and differential algebra [Ore33,Gau94,Gal85,BP96]. Formal power series with discounting have recently been further investigated in [ÉK05a,ÉK05b,Kui05,DR07].…”

Hyperexponential solutions of finite-rank ideals in orthogonal ore rings

“…which follows directly from Proposition 3.1 in [11, §8.3] or Theorem 9 in [9]. By the identity (7), an easy induction proves that φ is a ring homomorphism.…”

“…By Lemma 3 in [9] we conclude that θ i is K-pseudo-linear with respect to σ i and δ i , that is, θ i is additive and…”

“…. , n. The reader is referred to [9] for a general way to define the action of Ore polynomials on K. We need only a special kind of pseudo-linear operators. …”

“…Guided by the work of [9,10] we introduce the notion of orthogonal Ore algebras. These are general enough to include algebras of partial differential, shift, and q-shift operators over the field of multivariate rational functions but also simple enough to analyze the nonlinear compatibility conditions of a left ideal of rank one.…”