It is known that classical cotorsion pairs satisfy the “telescope conjecture for module categories”. We obtain a corresponding result in Gorenstein setting for the rings with finite Gorenstein weak global dimension. As applications, we obtain that the condition of rings such that the “first and second dimension conjectures” hold true can be relaxed from Gorenstein rings to left Noetherian rings with finite Gorenstein weak global dimension.
For d≥ 1 and s≥ 0 a (d,d+s) -graph is a graph whose degrees all lie in the interval {d, d+1, … ,d+s} . For r≥ 1 and a≥ 0 an (r,r+a) -factor of a graph G is a spanning (r,r+a) -subgraph of G. An (r,r+a) -factorization of a graph G is a decomposition of G into edge-disjoint (r,r+a) -factors. A graph is (r,r+a) -factorable if it has an (r,r+a) -factorization. Let σ (r,s,a,t) be the least integer such that if d≥σ (r,s,a,t) , then every (d,d+s) -simple graph G is (r,r+a) -factorable with x factors for at least t different values of x. In this paper we evaluate, or closely bound, σ (r,s,a,t) for all values of r,a,t≥ 1 and s≥ 0 . We also prove a number of results of which the following is a typical example. If a≥ 2 and r≥ 1 , then, when r is even and a is odd, every (d,d+s) -simple graph which is not an oddly-even graph has an (r,r+a) -factorization with x factors if and only if d+s/r+a< x ≤d/r. An oddly-even (d,d+s) -graph has |ℰ(G)| odd and has the degrees of all its vertices in {d,d+s} when d and d+s are both even. We prove similar statements as the example just given for all the other parities of r and a.
In this paper, we first demonstrate that every non-negative square matrix A with the spectral radius r(A)<1 has a matrix norm |||· ||| which is subordinate to a given monotone norm with respect to the partial ordering in 𝔐_m,1(ℝ) such that |||A|||<1. As an application, we establish a new Ćirić type fixed point theorem in ℝ^m -b-metric spaces with respect to the relation called “ maxco ”. Finally, we present an application to dynamic programming. We also present examples to illustrate the novelty of the results.
The aim of this work is to provide details how the well-known Banach’s contraction mapping principle (CMP) can be used to derive criteria for local asymptotic p-th mean stability of the trivial equilibrium of stochastic differential equations (SDEs) with infinite memory, driven by standard Wiener processes on the strong Banach space 𝕄_p of cádlág or cáglád stochastic processes with bounded p-th moments 𝔼[sup _-∞
