The concepts of free modules, projective modules, injective modules and the likeform an important area in module theory. The notion of free fuzzy modules was introducedby Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameriintroduced the concept of projective and injective L-modules. In this paper we give analternate definition for projective L-modules. We prove that e...

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus being weak approximation of rational points. We construct a 9-term Poitou–Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the ...

In a system of interlocking sequences, the assumption that three out of the four sequences are exact does not guarantee the exactness of the fourth. In [1], Hilton proved that, with the additional condition that it is differential at the crossing points, the fourth sequence is also exact. In this paper, we trace such a diagram and analyze the relation between the kernels and the images, in the ...

‎in the following text for arbitrary $x$ with at least two elements‎, ‎nonempty countable set $gamma$‎ ‎we make a comparative study on the collection of generalized shift dynamical systems like $(x^gamma,sigma_varphi)$ where $varphi:gammatogamma$ is an arbitrary self-map‎. ‎we pay attention to sub-systems and combinations of generalized shifts with counterexamples regarding devaney‎, ‎exact dev...

where Xs+1 is the fiber of gs. We get an exact couple of homotopy groups and a spectral sequence with E 1 = πt−s(Ks) and dr : E s,t r → Es+r,t+r−1 r . This spectral sequence converges to π∗(X) (where X = X0) if the homotopy inverse limit lim←Xs is contractible and certain lim 1 groups vanish. When X is connective, it is a first quadrant spectral sequence. For more background, see [Rav86]. In th...

This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number q, with minimal polynomial p, such that p(0) = −1, and where p has only one real root, then there exists a C(q), explicitly given here, such that: (1) For all > 0, all but finitely many integer quadratics P satisfy |P (q)| ≥ C(q)− H(P )2 where H is the he...