Volume 27 (2019), Issue 1

1. Lightlike hypersurfaces of an indefinite Kaehler manifold of a quasi-constant curvature

Dae Ho Jin ; Jae Won Lee.
We study lightlike hypersurfaces M of an indefinite Kaehler manifold M̅ of quasi-constant curvature subject to the condition that the characteristic vector field ζ of M̅ is tangent to M. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface M of M̅ such that (1) the screen distribution S(TM) is totally umbilical or (2) M is screen conformal.

2. [Retracted by the journal] α-modules and generalized submodules

Rafiquddin Rafiquddin ; Ayazul Hasan ; Mohammad Fareed Ahmad.
[This article has been retracted by the journal ; see Retractation notice https://doi.org/10.2478/cm-2019-0014]A QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω 1 we prove that an α-large submodule L of an ω 1-module M is summable if and only if M is summable.

3. Hilbert series of the Grassmannian and k-Narayana numbers

Lukas Braun.
We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the q-Hilbert series is a Vandermonde-like determinant. We show that the h-polynomial of the Grassmannian coincides with the k-Narayana polynomial. A simplified formula for the h-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k-Narayana numbers, i.e. the h-polynomial of the Grassmannian.

4. Generalized reverse derivations and commutativity of prime rings

Shuliang Huang.
Let R be a prime ring with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds: (i) F (xy) ± xy ∈ Z(R) (ii) F ([x, y]) ± [F (x), y] ∈ Z(R) (iii) F ([x, y]) ± [F (x), F (y)] ∈ Z(R) (iv) F (x ο y) ± F (x) ο F (y) ∈ Z(R) (v) [F (x), y] ± [x, F (y)] ∈ Z(R) (vi) F (x) ο y ± x ο F (y) ∈ Z(R) for all x, y ∈ I, then R is commutative.

5. On compatible linear connections of two-dimensional generalized Berwald manifolds: a classical approach

Csaba Vincze ; Tahere Reza Khoshdani ; Sareh Mehdi Zadeh Gilani ; Márk Oláh.
In the paper we characterize the two-dimensional generalized Berwald manifolds in terms of the classical setting of Finsler surfaces (Berwald frame, main scalar etc.). As an application we prove that if a Lands-berg surface is a generalized Berwald manifold then it must be a Berwald manifold. Especially, we reproduce Wagner’s original result in honor of the 75th anniversary of publishing his pioneering work about generalized Berwald manifolds.

6. On generalized derivations of partially ordered sets

Ahmed Y. Abdelwanis ; Abdelkarim Boua.
Let P be a poset and d be a derivation on P. In this research, the notion of generalized d-derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized d-derivations are introduced. The properties of the fixed points based on the generalized d-derivations are examined. The properties of ideals and operations related with generalized d-derivations are studied.