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Approximate property of a functional equation with a general involution

Bae J.-H., Park W.-G.

Demonstratio Mathematica

2018

Vol. 51, nr 1

304--308

In this paper, we prove the Hyers-Ulam stability of the functional equation f(x + y, z + w) + f(x + σ (y), z + τ(w)) = 2f(x, z) + 2f(y, w), where σ, τ are involutions.

On Jordan triple α-* centralizers of semiprime rings

Ashraf M., Mozumder M.R., Khan A.

Demonstratio Mathematica

2014

Vol. 47, nr 1

130--136

Let R be a 2-torsion free semiprime ring equipped with an involution *. An additive mapping T : R → R is called a left (resp. right) Jordan α-* centralizer associated with a function α : R → R if T(x2) = T(x)α(x*) (resp. T(x2) = α(x*)T(x)) holds for all x (…) R. If T is both left and right Jordan α-* centralizer of R, then it is called Jordan α-* centralizer of R. In the present paper it is shown that if α is an automorphism of R, and T : R → R is an additive mapping such that 2T(xyx) = T(x)α(y*x*) + α(x*y*)T(x) holds for all x; y (…) R, then T is a Jordan α-* centralizer of R.

Quantifiers on lattices with an antitone involution

Chajda I., Langer H.

Demonstratio Mathematica

2009

Vol. 42, nr 2

241-246

Quantifiers on lattices with an antitone involution are considered and it is proved that the poset of existential quantifiers is antiisomorphic to the poset of relatively complete sublattices.

About Various Methods of Calculating the Sum [formula]

Bryll G., Rygał G.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2006

Vol. 11

7--16

Pupils of secondary school as well as students often have problems with calculating the sums of the mth powers of successive natural numbers. In this paper we present certain methods of finding such sums.

A non-standard version of the Borsuk-Ulam theorem

Biasi C., Mattos D. de

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

111-119

E. Pannwitz showed in 1952 that for any n≥2, there exist continuous maps φ : Sn → Sn and f : Sn → R2 such that f(x) ≠ f(φ(x)) for any x ∈ Sn. We prove that, under certain conditions, given continuous maps ψ,φ : X → X and f : X → R2, although the existence of a point x ∈ X such that f(ψ(x)) = f(φ(x)) cannot always be assured, it is possible to establish an interesting relation between the points f(φψ(x)), f(φ2(x)) and f(ψ2(x)) when f(φ(x)) ≠ f(ψ(x)) for any x ∈ X, and a non-standard version of the Borsuk–Ulam theorem is obtained.