In this short comments, we will present some relations between the papers [2], [1] and [5], which may not discuss precisely in the paper [2]. The paper [2] studied the semigroup decay of the linearized Boltzmann equation in general Banach spaces in a torus, here the torus is size-dependent, which is based on the previous work [5] for fluid-nonfluid pointwise estimate. However, in paper [1], the...

Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the m-th Quillen K-group of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant. Let F be a number field, A its ring of integers and Km(A) the m-th Quillen K-group of A. It wa...

avila and molina [1] introduced the notion of generalized weak structures which naturally generalize minimal structures, generalized topologies and weak structures and the structures α(g),π(g),σ(g) and β(g). this work is a further investigation of generalized weak structures due to avila and molina. further we introduce the structures ro(g) and rc(g) and study the properties of the structur...