Finitely-hollow lifting and hollow-weak modules are two generalizations of hollow module which is a weaker version module. In this paper, we have defined another generalization hollow-lifting namely finitely-hollow-weak (f-hollow-weak lifting) Various properties been seen found that, it not closed under quotients summands in general, so sufficient conditions provided f-hollow-weak property inhe...

Let p be a strong type of an algebraically closed tuple over B = acl(B) in any theory T . Depending on a ternary relation |̂ ∗ satisfying some basic axioms (there is at least one such, namely the trivial independence in T ), the first homology group H∗ 1 (p) can be introduced, similarly to [3]. We show that there is a canonical surjective homomorphism from the Lascar group over B to H∗ 1 (p). We...

In this paper we investigate some relations between Bernstein polynomials and q-Euler numbers with weak weight α. From these relations, we derive some interesting identities on the q-Euler numbers with weak weight α. Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of p-adic rati...

Any charged particle moving faster than light through a medium emits Cherenkov radiation. We show that charged particles moving faster than light through the vacuum emit Cherenkov radiation. How can a particle move faster than light? The weak speed of a charged particle can exceed the speed of light. By definition, the weak velocity ^v&w is ^C f inuvuC in&/^C f inuC in&, where v is the velocity...

Any charged particle moving faster than light through a medium emits Cherenkov radiation. We show that charged particles moving faster than light through the vacuum emit Cherenkov radiation. How can a particle move faster than light? The weak speed of a charged particle can exceed the speed of light. By definition, the weak velocity v w is Ψ f in |v|Ψ in /Ψ f in |Ψ in , where v is the velocity ...

A general version of the fundamental theorem for System F is presented which can be instantiated to obtain proofs of weak βand βη-normalization and normalization by evaluation.