Bulletin of Iranian Mathematical Society
Volume:35 Issue: 1, 2009

One of the most important areas of investigation in number theory is the study of the distribution of rational or integral points on algebraic varieties. It is not a small historical miracle that an area of investigation in mathematics that started some 2000 years ago would still be of great interest. This field now has profound connections to various areasof modern mathematics including algebraic geometry, complex analysis, logic, and more recently harmonic analysis and automorphic representation theory. One of the central themes in the theory is to explore the relationship between geometric and arithmetic properties of algebraic varieties. One of the guiding principles is the idea that the roughgeometric classification of algebraic varieties according to the ampleness of the canonical (respectively anticanonical) line bundle should directly influence the arithmetic properties.

The purpose of this paper is to introduce and to discuss the concept of p-approximation and p-orthogonality in vector spaces, and to obtain some results on p- orthogonality in vector spaces similar to some well known results on the orthogonality in normed spaces. We also discuss the concept of p-extension of linear functionals on a vector space, and give a characterization of linear functionals on a subspace having a unique p-extension Hahn-Banach to the whole vector space.

Given a well-ordered semi-group with a minimal system of generators of ordinal type at most! n1 and of rational rank r, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with r variables such that the semi-group of the values of the polynomial ring is equal to. The construction uses a generalization of Favre and Jonsson’s version of MacLane’s sequence of key-polynomials [3].

Here, we develop the generalized frame theory. We introduce two methods for generating g-frames of a Hilbert space H. The first method uses bounded linear operators between Hilbert spaces. The second method uses bounded linear operators on `2to generate g-frames of H. We characterize all the bounded linear mappings that transform g-frames into other g-frames. We also characterize similar and unitary equivalent g-frames in term of the range of their linear analysis operators. Finally, we generalize thefundamental frame identity to g-frames and derive some new results.

Generalizing concepts “right Bezout” and “principal right ideal” of a ring R to modules, an R-module M is called n-epiretractable (resp. epi-retractable) if every n-generated submodule (resp. submodule) ofMR is a homomorphic image ofM. It is shown that if MR is finitely generated quasi-projective 1-epi-retractable, then EndR(M) is a right Bezout (resp. principal right ideal) ring if and only if MR is n-epi-retractable for all n  1 (resp. epiretractable). For a ring R and an infinite ordinal  |R|, the Rmodule M = F  N is epi-retractable where F is a free R-module with a basis set of cardinality and N is a -generated R-module with . A ring R is quasi Frobenius if every injective R-moduleis epi-retractable. Injective modules in R] are epi-retractable for every N 2 R] if and only if every non-zero factor ring of S is a quasi Frobenius ring where S is an endomorphism ring of a progenerator in R]