The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...

Let M be a simple 3-manifold with F a component of ∂M of genus at least two. For a slope α on F , we denote by M(α) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of α on F . Suppose that α and β are two separating slopes on F such that M(α) and M(β) are reducible. Then the distance between α and β is at most 2. As a corollary, if g(F ) = 2, then there is at mos...

In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fields. We also prove that an L-reducible Finsler metric admitting a concurrent vector field reduces to a Landsberg metric.In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α,β) -metric admits no concurrent vector fi...

1.1 The Quantifiable Continuum of Uncertainty. Notice that once uncertainties are acknowledged as unquantified uncertainties, increased knowledge about the uncertainties will determine their position on the continuum.. .. .. 11 1.2 Typology for sources and degree of uncertainty. Adapted from Van Asselt's (2000) proposed typology for uncertainties in integrated assessment.. .. 12 1.3 Some potent...