An Engel condition with skew derivations

Let R be a prime ring and set [x, y]₁ = [x, y] = xy ? yx for ${x,y\in R}$ and inductively [x, y] _k = [[x, y] _k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] _k  = 0 for all ${x \in L}$ , where k is a fixed positive integer, then charR = 2 and ${R\subseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

The group of commutativity preserving maps on strictly upper triangular matrices

Let N = N _n (R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ? φ(x)φ(y) = φ(y)φ(x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of N can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.     

Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfying

δ([x,y])=[δ(x),y]+[x,δ(y)]     

Identities with Generalized Derivations in Prime Rings

In this paper we investigate identities with two generalized derivations in prime rings. We prove, for example, the following result. Let R be a prime ring of characteristic different from two and let F ₁, F ₂ : R → R be generalized derivations satisfying the relation F ₁(x)F ₂(x) + F ₂(x)F ₁(x) = 0 for all ${x \in R}$ . In this case either F ₁ = 0 or F ₂ = 0.     

Strong commutativity preserving maps on Lie ideals

Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ²=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4.     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Numerical solution of a class of random boundary value problems

This paper deals with the nonlinear two point boundary value problem y″ = f(x, y, y′, R₁,…, R_n), x₀ < x < x_fS₁y(x₀) + S₂y′(x₀) = S₃, S₄y(x_f) + S₅y′(x_f) = S₆ where R₁,…, R_n, S₁,…, S₆ are bounded continuous random variables. An approximate probability distribution function for y(x) is constructed by numerical integration of a set of related deterministic problems. Two distinct methods are described, and in each case convergence of the approximate distribution function to the actual distribution function is established. Primary attention is placed on problems with two random variables, but various generalizations are noted. As an example, a nonlinear one-dimensional heat conduction problem containing one or two random variables is studied in some detail.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

Optimal Transportation with an Oscillation-Type Cost: The One-Dimensional Case

The main result of this paper is the existence of an optimal transport map T between two given measures μ and ν, for a cost which considers the maximal oscillation of T at scale δ, given by ω _δ (T) :?=??sup_|x???y|?<?δ |T(x)???T(y)|. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.