A criterion for the existence of a birational embedding into a projective plane with three collinear Galois points for algebraic curves is presented. The extendability of an automorphism induced by a Galois point to a linear transformation of the projective plane is also discussed, under the assumption that two Galois points exist.

In this paper, we prove the equivalence between a $K$-functionals constructed by the Sobolev space corresponding to the Laplace Cherednik-Opdam operator and a modulus of smoothness for the Cherednik-Opdam transform.

In this paper, inspired by Chen-Köbis-Köbis-Yao, we investigate the ordinal structure of the weighted set relation in the framework of set optimization problem. Using characterization theorems of set relations via nonlinear scalarization technique, we give relationships between traditional set relations and the weighted set relations.

A finite rank difference of two composition operators is studied on an analytic Banach space $\mathcal{B}$ on a domain $\mathcal{D}$ of the complex plane. It is shown that the rank is either zero or one when $\mathcal{D}$ is a bounded domain and $\mathcal{B}$ contains all bounded analytic functions on $\mathcal{D}$.

We show that any open ball centered at the origin of a complex inner product space endowed with a slightly modified Möbius addition is a gyrocommutative gyrogroup. A Möbius scalar multiplication by complex numbers can be naturally introduced so that some of the axioms of real inner product gyrovector spaces are satisfied. Moreover, the modified Möbius addition on the open ball of a complex inner product space can be characterized by three fundamental requirements.