For a meromorphic function $f$ in the complex plane, we shall introduce the definition of five-value rich line of $f$, and study the uniqueness of meromorphic functions of finite order in an angular domain by involving the five-value rich line and Borel directions. Finally, the relationship between a five-value rich line and a Borel direction is discussed, that is, every Borel direction of $f$ ...


 In this paper, we shall prove Theorem 1
 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ a proximate of and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists $\phi_0$ $0\le \phi_0 < 2\pi$ such that the inequality
 \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))...

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along ...

Define z to be the smallest cardinality of a function f : X → Y withX, Y ⊆ 2 such that there is no Borel function g ⊇ f . In this paper we prove that it is relatively consistent with ZFC to have b < z where b is, as usual, smallest cardinality of an unbounded family in ω. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists X ⊆ 2 s...

M. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kähler manifold preserved by the complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, L. O’Shea and R. Sjamaar proved a convexity result for the moment...

A perturbation involving a small parameter of the Black-Scholes-Merton model with time dependent volatility is considered. A pricing formula is derived as an asymptotic series in powers of the small parameter. The summation of this series is performed, using methods of the theory of Borel summability in a suitable direction.