We give a simple proof of the duality theorem for the Monge-Kantorovich problem in the Euclidean setting. The selection lemma which is useful in the theory of stochastic optimal controls plays a crucial role.

It is shown that there exist infinitely many PSL(2,5) sextic fields with a power basis.

G. Brosch improved the theorem of Nevanlinna for four values theorem and proved that let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ CM, and let a and b be two finite complex numbers such that $a,b\not\in \{0,1\}$. If $f=a\Leftrightarrow g=b$, then f is a fractional linear transformation of g. In this paper we extend this theorem by using the idea of weighted sharing.

Is the corank an invariant of the blow-analytic equivalence between real analytic function germs? We give a positive answer in the case of the blow-Nash equivalence, which is a natural variant of the blow-analytic equivalence for Nash function germs. In addition, we prove that the index is also a blow-Nash invariant. Their proof is based on the computation of some virtual Poincaré polynomials and zeta functions associated to a Nash function germ.

We prove a uniqueness theorem for meromorphic functions sharing a single value with unit weight which improves a recent result of A. H. H. Al-Khaladi.

In [1], R. Fehlmann and F. P. Gardiner studied an extremal problem for a finite Riemann surface to establish a slit mapping theorem. In this article, we give a condition for non-uniqueness of such slit mappings, by using a deformation of a Riemann surface.

We consider a class of lightlike hypersurfaces M, of semi-Riemannian manifolds, whose shape operator is conformal to the shape operator of its screen distributions. Integrability and characterization results are then established.

We derive the exact distributions of R = X + Y, U = XY and W = X/(X + Y) and the corresponding moment properties when X and Y follow the generalized bivariate Pareto distribution. The expressions turn out to involve special functions.

An important problem in the theory of Lagrangian submanifolds is to find non-trivial examples of Lagrangian submanifolds in complex Euclidean spaces with some given special geometric properties. In this article, we provide a new method to construct Lagrangian surfaces in the complex Euclidean plane C² by using Legendre curves in S³(1) $\subset$ C². We also investigate intrinsic and extrinsic geometric properties of the Lagrangian surfaces in C² obtained by applying our construction method. As an application we provide some new families of Hamiltonian minimal Lagrangian surfaces in C² via our construction method.

A nondegenerate CR manifold has a natural conformal class consisting of hermitian forms, which are called Levi forms, on the hyperdistribution. It is important to study the objects on CR manifolds which are independent of the choice of Levi forms. A CR Einstein-Weyl structure defined in [8] is one of such objects. Our purpose is to find nondegenerate CR manifolds with CR Einstein-Weyl structures.

In this paper, we shall obtain many examples of CR Einstein-Weyl manifolds. These examples are actually derived from the fact that CR Einstein-Weyl structures are closely concerned with connections of the canonical bundle of CR manifolds.

Let V_δ denote a local level surface for function-germ f:(Rⁿ⁺¹,0) → (R,0). A mapping degree formula for difference of the Euler characteristics of V_δ$\cap$ {g ≤ 0} and V_δ$\cap$ {g ≥ 0} is given, when level surfaces of a function g:(Rⁿ⁺¹,0) → (R,0) are parallelizable.