Let U = Tri(A,M,B) be a triangular algebra, where A, B are unital algebras over a field F and M is a faithful (A,B)-bimodule. Assume that ξ ∈ F and L : U → U is a map. It is shown that, under some mild conditions, L is linear and satisfies L([X, Y ]) = [L(X), Y ] + [X,L(Y )] for any X,Y ∈ U with [X, Y ] = XY − Y X = 0 if and only if L(X) = φ(X) + ZX + f(X) for all A, where φ is a linear derivat...

A classical theorem of S. Bochner states that a function f : Rn → C is the Fourier transform of a finite Borel measure if and only if f is positive definite. In 1938, I. Schoenberg found a beautiful complement to Bochner’s theorem. We present a non-technical derivation of of Schoenberg’s theorem that relies chiefly on the de Finetti theorem and the law of large numbers of classical probability ...

In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L(X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D : A(X) → L(X) satisfying the relation D(Am+n) = D(Am)An + AmD(An) for all A ∈ A(X) and some fixed integers m ≥ 1, ...

We present the technique of derivation of a theory to obtain an (n + 1)f -degrees-of-freedom theory from an f -degrees-of-freedom theory and show that one can calculate all of the quantities of the derived theory from those of the original one. Specifically, we show that one can use this technique to construct, from an integrable system, other integrable systems with more degrees of freedom.

Model theory, a branch of mathematical logic which involves the study of sets definable in a mathematical structure using an appropriate first order language, has recently shown its capacity to produce applications to other domains of mathematics, principally to algebra, algebraic geometry and number theory. The main example is the proof by Ehud Hrushovski of a conjecture of algebraic geometry,...

Recent models in distributional semantics consider derivational patterns (e.g., use → use + f ul ) as the result of a compositional process, where base term and affix are combined. We exploit such models for German particle verbs (PVs), and focus on the task of learning a mapping function between base verbs and particle verbs. Our models apply particle-verb motivated training-space restrictions...

In the theory of rings and near-rings, the properties of derivations are an important topic to study, see [2, 3, 7, 10]. In [6], Jun and Xin applied the notions in rings and nearrings theory to BCI-algebras, and obtained some related properties. In this paper, the notion of left-right (resp., right-left) f -derivation of a BCI-algebra is introduced, and some related properties are investigated....

i=1 t i) k (87) We now comment on how the derivation changes when n = 2 and k 3. The case n 3 and k = 2 follows as well due to the symmetric functional equation (Equation 12). Note that up to Eq. 57 the derivation is valid when n = 2. Furthermore, note that the sum in Eq. 56 consists now of one term where l = j 1 = 1. Thus, Eq. 56 and Eq. 57 yield, using x i = z ij1 y j1 + z in y n (n=2, j 1 = ...

The notion of circular security of pseudorandom functions (PRF) was introduced in Distance Bounding Protocols. So far, only a construction based on a random oracle model was proposed. Circular security stands between two new notions which we call Key Dependent Feedback (KDF) security and Leak security. So far, only a construction based on a random oracle was proposed. We give an algebraic const...

Several authors 1–4 have studied derivations in rings and near rings. Jun and Xin 5 applied the notion of derivation in ring and near-ring theory to BCI-algebras, and as a result they introduced a new concept, called a regular derivation, in BCI-algebras. Zhan and Liu 6 studied f-derivations in BCI-algebras. Alshehri 7 applied the notion of derivations to incline algebras. Alshehri et al. 8 int...