International Journal of Mathematics

In this paper, we prove that two Randers metrics are pointwise projectively related if and only if they have the same Douglas tensors and the corresponding Riemannian metrics are projectively related. Moreover, Randers metrics of constant flag curvature and Einstein–Randers metrics are considered.

We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to prove that covariant representations of product systems admit isometric dilations.

In this paper, we study the parameter space of the quadratic polynomial family f_λ,μ(z, w) = (λz + w², μw + z²), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family f_λ,μ, we prove that the sum of the Lyapunov exponents is continuous.

A quadrilateral of factors is an irreducible inclusion of factors N ⊂ M with intermediate subfactors P and Q such that P and Q generate M and the intersection of P and Q is N. We investigate the structure of a noncommuting quadrilateral of factors with all the elementary inclusions P ⊂ M, Q ⊂ M, N ⊂ P, and N ⊂ Q 2-supertransitive. In particular, we classify noncommuting quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between P and Q for quadrilaterals coming from α-induction and asymptotic inclusions.