No abstract uploaded!

No abstract uploaded!

We prove a second fundamental theorem in the sense of Nevanlinna's theory of meromorphic functions replacing the constants$a$in $$\bar N$$ ($r, f, a$) by rational functions$R$with$R(∞)=a$. The key argument is Ahlfors's second fundamental theorem from his theory of covering surfaces.

Let$X$be a compact Hausdorff space. A kernel function on$X$×$X$, enjoying additional properties, naturally defines a semi-inner product structure on certain subspaces of all finite signed Borel measures on$X$. This paper discusses the question of completeness of such spaces.

The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established.