The following result is proved: Let R be a 2-torsion free semiprime ring, and let T : R → R be an additive mapping, related to a surjective homomorphism θ : R → R, such that 2T (x2) = T (x)θ(x) + θ(x)T (x) for all x ∈ R. Then T is both a left and a right θ-centralizer.

The main result: Let R be a 2-torsion free semiprime ring and let T : R → R be an additive mapping. Suppose that T (xyx) = xT (y)x holds for all x, y ∈ R. In this case T is a centralizer.

In 1940, Ulam [13] proposed the Ulam stability problem of additive mappings. In the next year, Hyers [5] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L is the unique additive mapping s...

A new approach of reliable image and video transmission over noisy channel is proposed. The proposed subband modulation scheme combines source coding and channel modulation scheme t o achieve high compression ef ic iency and preferable quality. Further performance gain is obtained by multiresolution modulat ion and bits re-mapping scheme that assign e f i c i en t mapping f rom source codeword ...

Let X,Y be Banach modules over a C∗-algebra and let r1, . . . , rn ∈ R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: ∑n j 1 f −rjxj ∑ 1≤i≤n,i / j rixi 2 ∑n i 1 rif xi nf ∑n i 1 rixi . We show that if ∑n i 1 ri / 0, ri, rj / 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation...

The problem considered in the paper can be described as follows. We are given a continuous mapping from one metric space into another which is regular (in the sense of metric regularity or, equivalently, controllability at a linear rate) near a certain point. How small may be an additive perturbation of the mapping which destroys regularity? The paper contains a new proof of a recent theorem of...

Recently the generalizedHyers-Ulam orHyers-Ulam-Rassias stability of the following functional equation ∑m j 1 f −rjxj ∑ 1≤i≤m,i / j rixi 2 ∑m i 1 rif xi mf ∑m i 1 rixi where r1, . . . , rm ∈ R, proved in Banach modules over a unital C∗-algebra. It was shown that if ∑m i 1 ri / 0, ri, rj / 0 for some 1 ≤ i < j ≤ m and a mapping f : X → Y satisfies the above mentioned functional equation then the...

A new concept of pre-semicompact L -subsets is introduced, where L is a fuzzy lattice. The pre-semicompact L -subset is semi-precompact, hence it is also semicompact and strongly compact. It is described with collection of pre-semiclosed L -subset, constant α -net and α -level filter base. It is a good L -extension; it is hereditary for pre-semiclosed L -subset; it is finitely additive; it is i...

This review highlights the key role of modularity and the additive factors method in functional neuroimaging. Our focus is on structure-function mappings in the human brain and how these are disclosed by brain mapping. We describe how modularity of processing (and possibly processes) was a key point of reference for establishing functional segregation as a principle of brain organization. Furth...