In this paper we consider some (a, b)- metrics such as generalized Kropina, Matsumoto and F=(a+b)2/a metrics, and obtain necessary and sufficient conditions for them to be Einstein metrics when b is a constant Killing form. Then we prove with this assumption that the mentioned Einstein metrics must be Riemannian or Ricci flat.

We study in a unitary way the SCHUR-convexity or concavity of the Stolarsky and Gini means Da,b (x, y) and Sa,b (x, y), for fixed x, y>0, x¹y.

This paper is concerned with bounded real criterion for linear neutral delay systems. Two new delay-dependent bounded real LEMMAs (BRLs) are obtained in this paper, in which, Lyapunov theory is used to derive the first delay-dependent representation for BRL. Using a descriptor model transformation of the system and a new Lyapunov-Krasovskii functional, a less conservative bounded real LEMMA is obtained compared to that of the first BRL. Then sufficient conditions for the system to possess an H¥-norm less than a prescribed level, is given in terms of a linear matrix inequality (LMI). The significant advantage of the derived bounded real LEMMAs is their efficiency in designing H¥ controller for the closed-loop neutral systems when delayed term coefficients depend on the controller parameters. Numerical examples are given which illustrate the effectiveness of our proposed BRLs.

We exhibit an explicit construction for the second cohomology group H2 (L, A) for a Lie ring L and a trivial L-module A. We show how the elements of H2 (L, A) correspond one-to-one to the equivalence classes of central extensions of L by A, where A now is considered as an abelian Lie ring.For a finite Lie ring L we also show that H2 (L, C*) @ M (L), where M (L) denotes the SCHUR multiplier of L. These results match precisely the analogue situation in group theory.

A celebrated result of I. SCHUR asserts that the derived subgroup of a group is finite provided the group modulo its center is finite, a result that has been the source of many investigations within the Theory of Groups. In this paper we exhibit a similar result to SCHUR’s Theorem for vector spaces, acted upon by certain groups. The proof of this analogous result depends on the characteristic of the underlying field. We also give linear versions of corresponding theorems of R. Baer and P. Hall.

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