Border bases and kernels of homomorphisms and of derivations

Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.     

Weak-local derivations and homomorphisms on C*-algebras

We prove that every weak-local derivation on a C*-algebra is continuous, and the same conclusion remains valid for weak*-local derivations on von Neumann algebras. We further show that weak-local derivations on C*-algebras and weak*-local derivations on von Neumann algebras are derivations. We also study the connections between bilocal derivations and bilocal*-automorphism with our notions of extreme-strong-local derivations and automorphisms.     

On Lie ideals with derivations as homomorphisms and anti-homomorphisms

In [2, Theorem 3], Bell and Kappe proved that if d is a derivation of a prime ring R which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal I of R, then d = 0 on R. In the present paper our objective is to extend this result to Lie ideals. The following result is proved: Let R be a 2-torsion free prime ring and U a nonzero Lie ideal of R such that u ² ∈ U, for all u ∈ U. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then either d=0 or U ?Z(R).     

Fixed points and functional equations connected with derivations on Banach algebras

Let 1≦m≦4 be a fixed integer and let f:X→Y be a mapping with X, Y two vector spaces. The functional equation (1.1) is said to be additive if m=1, quadratic if m=2, cubic if m=3 and quartic if m=4, respectively. For convenience, a solution of (1.1) will be called an m-mapping. Let $\mathcal{A}$ , $\mathcal{B}$ be two algebras. An m-mapping $f\colon \mathcal{A} \to \mathcal{B}$ will be called m-derivation if the equality f(xy)=x ^m f(y)+f(x)y ^m is fulfilled for all $x,y\in \mathcal{A}$ . In this paper, we use a fixed point method to prove the stability and hyperstability of m-derivations on Banach algebras.     

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.     

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.     

A fixed point method for almost partial derivations of Jensen type n-variable functional equations

In this paper, we will apply a fixed point method for proving the generalized Hyers–Ulam–Rassias stability of the partial derivations for Jensen type n-variable functional equations.     

Commutation relations,homomorphisms, and differential equations

We present an algebraic approach to the theory of ordinary differential equations and indicate a method for constructing first integrals of such equations.     

On derivations arising in differential equations

Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation.     

Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W_p(m_i,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Q_k(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images ρ_k(Σ⁺) of the Moore-Penrose inverse Σ⁺ of Σ are mutually orthogonal: ρ_k(Σ⁺)ρ_?(Σ⁺)=0 for k≠?.     

Approximate linear derivations and functional inequalities with applications

In this paper, we prove that any approximate linear derivation on a semisimple Banach algebra is continuous. We deal with the functional inequalities associated with additive mappings and some stability theorems are proved. Based on these facts, we obtain some results for the functional inequalities corresponding to the additive mappings and the equation $f\left(xy\right)=xf\left(y\right)+f\left(x\right)y$.     

Characterization of groups of involutions by means of composite functional equations in two variables

Acta Mathematica Hungarica - The groups all of whose elements are involutions are characterized by means of two equivalent composite functional equations in two variables.     

Independence and graph homomorphisms graph homomorphisms

A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices. This is best possible. The proof proceeds by producing a homomorphism to the 7-cycle and invoking the No Homomorphism Lemma. For k ≥ 4, a graph with n vertices, odd girth 2k+1, and minimum degree exceeding n/(k+1) contains an independent set with at least kn/(2k+1) vertices; however, we suspect this is not best possible. © 1993 John Wiley & Sons, Inc.