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WOS: 000406745600023, Let R be a semiprime ring and L is a Lie ideal of R such that L 6 not subset of Z(R) A map F : R -> R is called a multiplicative generalized derivation if there exists a map d : R -> R such that F(xy) = F(x)y + x d(y), for all x, y is an element of R . In the present paper, we shall prove that d is a commuting map on L if any one of the following holds: i) F(uv) = +/- uv, ii) F(u v) = +/- vu, iii) F(u) F(v) = -/+ uv, iv) F(u) F(v) = +/- vu, v) F(u) F(v) +/- uv is an element of Z, vi) F(u) F(v) +/- vu is an element of Z, vii) [F(u), v] +/- [u, G(v)] = 0; for all u, v is an element of L

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Moreover, we provide its generalization in continuous setting. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.

In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-Sandor means in terms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharp inequalities involving Neuman-Sandor, contra-harmonic, root-square, and other means of two positive real numbers.

WOS: 000338514500029 In this paper, we study modules with the properties (CRE) and (CREE), which are adapted Zoschinger's modules with the properties (E) and (EE). It is shown that: (1) a module M has the propery (CREE) if and only if every submodule of M has the propery (CRE); (2) a ring R is Rad-supplemented if and only if every left R-module has the propery (CRE); (3) over a commutative Von Neumann regular ring a module M has the propery (CRE) if and only if M is cofinitely injective.

In this paper, we obtain the necessary and sufficient conditions for having the maximum principle and existence of positive solutions for some cooperative systems involving Schrödinger operators defined on unbounded domains. Then, we deduce the existence of solutions for semi-linear systems. Finally we discuss the generalized maximum principle (gf q-positivity) for non cooperative systems.