In this paper we study the properties of special (α, β)-metric α α−β + β, the Randers change of Matsumoto metric. We find a necessary and sufficient condition for this metric to be of locally projectively flat and we prove the conditions for this metric to be of Berwald and Douglas type.

This paper is mainly concerned with scrambled sets for Luroth map. It shown that all have null Lebesgue measure and there exists a set full Hausdorff dimension.

David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain Ω in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of Ω, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance f...

we prove that every r-quadratic metric of scalar flag curvature with a dimension greater than twois of constant flag curvature. then we show that generalized douglas-weyl metrics contain r-quadraticmetrics as a special case, but the class of r-quadratic metric is not closed under projective transformations