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Iterated quasi-arithmetic mean-type mappings

Paweł Pasteczka (2016)

Colloquium Mathematicae

We work with a fixed N-tuple of quasi-arithmetic means M ₁ , . . . , M N generated by an N-tuple of continuous monotone functions f ₁ , . . . , f N : I → ℝ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping I N ∋ b ↦ ( M ₁ ( b ) , . . . , M N ( b ) ) tend pointwise to a mapping having values on the diagonal of I N . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means M ₁ , . . . , M N taken on b. We effectively...

Jordan *-derivation pairs on standard operator algebras and related results

Dilian Yang (2005)

Colloquium Mathematicae

Motivated by Problem 2 in [2], Jordan *-derivation pairs and n-Jordan *-mappings are studied. From the results on these mappings, an affirmative answer to Problem 2 in [2] is given when E = F in (1) or when 𝓐 is unital. For the general case, we prove that every Jordan *-derivation pair is automatically real-linear. Furthermore, a characterization of a non-normal prime *-ring under some mild assumptions and a representation theorem for quasi-quadratic functionals are provided.

Kurepa's functional equation on semigroups.

Bruce R. Ebanks (1982)

Stochastica

The functional equation to which the title refers is:F(x,y) + F(xy,z) = F(x,yz) + F(y,z),where x, y and z are in a commutative semigroup S and F: S x S --> X with (X,+) a divisible abelian group (Divisibility means that for any y belonging to X and natural number n there exists a (unique) solution x belonging to X to nx = y).

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