Rational twisted power series over a (commutative) filed are studied. We give several characterizations of such series which are similar to the classical results concerning rational power series over a commutative field. In particular, we prove a version of Kronecker's lemma for the rationality of twisted power series.

No abstract

We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.

A version of the Penrose transform is introduced in the split signature. It relates the cohomological data on CP 3 \ RP 3 and kernel of differential operators on M , the (real) Grassmannian of 2-planes in R 4 . As an example we derive the following cohomological interpretation of the so-called X-ray transformwhere Γ ω (M, ε[−1]) and Γ ω (M, ε[−3]) are real analytic sections of certain (homogeneous) line bundles on M , c stands for cohomology with compact support and 2,2 is the ultrahyperbolic operator. Furthermore, this gives a cohomological realization of the so-called "minimal" representation of SL(4, R). We also present the split Penrose transform in split instanton backgrounds.

Abstract. Let F be an algebraically closed field and T : Mn(F ) −→ Mn(F ) be a linear transformation. In this paper we show that if T preserves at least one eigenvalue of each matrix, then T preserves all eigenvalues of each matrix. Moreover, for any infinite field F (not necessarily algebraically closed) we prove that if T : Mn(F ) −→ Mn(F ) is a linear transformation and for any A ∈ Mn(F ) with at least an eigenvalue in F , A and T (A) have at least one common eigenvalue in F , then T preserves the characteristic polynomial.