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Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and U=[AMNB] is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero. [ABSTRACT FROM AUTHOR]

Let ( T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω( x, f) and P( f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω( x, f) = { y| there exist a sequence of points x ∈ T and a sequence of positive integers n < n < ··· such that lim x = x and lim $$f^{n_{k}}$$ ( x ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω( x, f) = Ω( x, f) for any x ∈ T. (3) ∩ f ( T) = P( f), and ω( x, f) is a periodic orbit for every x ∈ T and map h: x → ω( x, f) ( x ∈ T) is continuous. (4) Ω( x, f) is a periodic orbit for any x ∈ T. [ABSTRACT FROM AUTHOR]