let a be a unital algebra over a field of characteristic zero. we show that every derivation from( ) n m a into its dual ( ) n m a ∗ is the sum of an inner derivation and a derivation induced by a derivationfrom a into a∗

The existence of linear differential resolvents for z for any root z of an ordinary polynomial with coefficients in a given ordinary differential field has been established, where α is an indeterminate constant with respect to the derivation of the given field. In this paper we consider several alphas. We will call a finite sum of indeterminate powers of a variable v a pseudopolynomial in v . W...

The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F . This paper presents an expository proof of this fact modeled on a similar equivalence from polynomial Galois theory, whose...

Abstract. Let F be a field of characteristic not 2 and assume all algebras are over F. We establish several conjugacy theorems for the special linear Lie algebra sl2 over an F -algebra which is a UFD. We find the structure of the full automorphism group as well as conditions for when two such algebras are isomorphic. We also study the structure of their derivation algebras. Often we work with m...

In this paper, we investigate simple eco-grammar systems with n agents. The number of agents which are active at each derivation step depends on the number of steps which have already been carried out since the beginning of the development. This dependency is expressed by a function f . For each pair of n and f , corresponding language families are defined. These families are compared with each...

Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C〈Y1, . . . , Y5〉, with Y1, . . . , Y5 differentially independent over C. We show that there is a Picard-Vessiot extension E ⊃ F for a matrix equation X ′ = XA(Yi), with differential Galois group SO3, with the property that if F is any differential field with field of constants C then...

F (qiM) (1− qiM) . Since |U/{k}| = k − 1 , the expected size of Vk,U/{k} is F (qiM) (1 − qiM) . Note that, for F large enough, the actual realization of random number of bits in Vk,S/{k} is in the interval F (qiM) (1 − qiM) K−k+1 ± o(F ), with high probability. For simplicity, the o(F ) term is ignored in the following derivation. Consider a fixed value of s in Line 7 and a fixed subset S of ca...

Let R be a prime ring with a right generalized (α, β)derivation f and let a ∈ R. Suppose that af(x)n = 0 for all x ∈ R, where n is a fixed positive integer. Then af(x) = 0 for all x ∈R. In particular, if f is either a regular right generalized (α, β)-derivation or a nonzero generalized (α, β)-derivation, then a = 0. In [13] I. N. Herstein proved that if R is a prime ring and d is an inner deriv...

Artemov’s logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidence-based approach to reasoning about knowledge. Additional atoms in LP have form t : F , read as “t is a proof of F” (or, more generally, as “t is an evidence for F”) for an appropriate system of terms t called proof polynomials. In this paper, we answer two well-know...

Artemov’s logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidence-based approach to reasoning about knowledge. Additional atoms in LP have form t : F , read as “t is a proof of F” (or, more generally, as “t is an evidence for F”) for an appropriate system of terms t called proof polynomials. In this paper, we answer two well-know...