In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of...

In these notes we will first define projective modules and prove some standard properties of those modules. Then we will classify finitely generated projective modules over Dedekind domains Remark 0.1. All rings will be commutative with 1. 1. Projective modules Definition 1.1. Let R be a ring and let M be an R-module. Then M is called projective if for all surjections p : N → N ′ and a map f : ...

Given two Banach spaces X and Y, we analyze when the projective tensor product X?ˆ?Y has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a compactly generated (WCG) space Hilbert space. For instance, show that: (i) WLD if only both Y are all operators from to Y? X? have separable range; (ii) WCG same holds for under assumption that every operator compact; (iii) ?p(?)?ˆ...

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components. 1 Background definitions and main results We say that a projective plane is transitive (respectively primitive) if it ...

in this paper $s$ is a monoid with a left zero and $a_s$ (or $a$) is a unitary right $s$-act. it is shown that a monoid $s$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $s$-act is quasi-projective. also it is shown that if every right $s$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang (2012) for unique complete Kähler-Einstein metric Cheng Yau (1980), Kobayashi (1984), Tian (1987) Bando (1990) on quasi-projective manifolds. The main tools are solution second-order ordinary differential equations (ODEs) with constant coefficients spectral theory Laplacian operator a closed manifold.