I recall the well-known sufficient conditions for the bound r < 16ǫ on the spectrum of the primordial tensor perturbation. Two recent papers claim a violation of this bound, without stating explicitely any violation of the sufficient conditions. The following conditions are sufficient for the bound r < 16ǫ on the primordial tensor perturbation. • Inflation is of the slow-roll variety. • The cur...

All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith’s theorem stating conditions ensuring that the sum and ...

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring. It originally used to solve longstanding conjecture Borodin, stating that every planar graph without cycles lengths 4 8 is 3-choosable. In this paper, we give three sufficient conditions for be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 1.7) are stated in ``color extendability'' form, u...

one of the most common and appealing issues grabbing the attention of the critics of contemporary arabic literature is innuendo and its applications which are not much attended to by persian speakers. in this paper we have focused on innuendo and its applications. the questions we have raised and have attempted to respond to include: what is niq?b (innuendo) and where do its roots lie? what is ...