This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.

In this paper, we construct some rotationally symmetric pseudo-K?hler-Einstein metrics on various holomorphic line bundles over projective spaces and their disc bundles. We also discuss the phase change phenomenon when one suitably changes parameters.

In this paper, we will study the operator given by F(z)=(f(z1)+f′(z1)P(z0),(f′(z1))1/kz0T)T, where z=(z1,zT0)T belongs to the unit ball Bⁿ in ?n, z1∈U=B1, z0=(z2,?,zn)T∈?n-1, and P:?n-1→? is a homogeneous polynomial of degree k (k≥2), the holomorphic branch is chosen such that (f′(0))1/k=1. We will give different conditions for P such that the modified operator preserves the properties of almost spirallikeness of type β and order α, spirallikeness of type β and order α, and strongly spirallikeness of type β and order α, respectively.

This paper presents a weighted L² estimate with power weights for the maximal operator of commutators generated by compactly supported multipliers and Lipschitz functions. As an application, we study the almost convergence of the commutators, which is generated by the Bochner-Riesz means under the critical index and Lipschitz functions, for functions in L^p (p≥2).

In this paper, we first show the uniqueness of invariant measures for the stochastic fast diffusion equation, which follows from an obtained new decay estimate. Then we establish the Harnack inequality for the stochastic fast diffusion equation with nonlinear perturbation in the drift and derive the heat kernel estimate and ultrabounded property for the associated transition semigroup. Moreover, the exponential ergodicity and the existence of a spectral gap are also investigated.

This paper deals with a modified nonlinear inexact Uzawa (MNIU) method for solving the stabilized saddle point problem. The modified Uzawa method is an inexact inner-outer iteration with a variable relaxation parameter and has been discussed in the literature for uniform inner accuracy. This paper focuses on the general case when the accuracy of inner iteration can be variable and the convergence of MNIU with variable inner accuracy, based on a simple energy norm. Sufficient conditions for the convergence of MNIU are proposed. The convergence analysis not only greatly improves the existing convergence results for uniform inner accuracy in the literature, but also extends the convergence to the variable inner accuracy that has not been touched in literature. Numerical experiments are given to show the efficiency of the MNIU algorithm.

In this paper, under the first-order moment condition of the infinitely divisible distribution on Gel’fand triple, we use Riesz potential to construct fractional Lévy random fields on Gel’fand triple by white noise approach. We investigate the distribution and sample properties of isotropic and anisotropic fractional Lévy random fields, respectively.

In this paper, we study the weighted norm inequalities for commutators formed by a class of one-sided oscillatory integral operators and functions in one-sided BMO spaces.

In this paper, we obtain some boundedness on the weighted Lebesgue spaces and the weighted Hardy spaces for the parametrized Littlewood-Paley operators with the kernel Ω satisfying the logarithmic type Lipschitz conditions.

In this paper, we consider a conformal minimal immersion f from S² into a hyperquadric Q₂, and prove that its Gaussian curvature K and normal curvature K^⊥ satisfy K + K^⊥ = 4. We also show that the ellipse of curvature is a circle.

Based on the theory of semi-global C² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.

In this paper, we use the elementary and analytic methods to study the computational problem of one kind mean value involving the classical Dedekind sums and two-term exponential sums, and give two exact computational formulae for them.