Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed.The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements.The method is made invar-iant domain preserving for the Euler equations using convex limiting and is tested on vari-ous benchmarks.

Completely Invariant Domains of Holomorphic Self-Maps on C

Let/(z) be a holomorph.self-map on C.-G-(0) with essential singularities 0 and It is proved that f(z) has a completdy invariant domain.D.F(f),then D is doubly connected and D contains all the singularities of the inverse of f(z),moreover,if f is of the finite type, then D=F(f). This result implies that f(z) has at most one completely invariant domain in F(f).

The degree of biholomorphic mappings between special domains in C^n preserving 0

Let Gi be a closed Lie subgroup of U（n）, Ωi be a bounded Gi-invariant domain in Cn which contains 0, and （9（Cn）Gi = C, for i= 1,2. If f ： f21 →2 is abiholomorphism, and f（0） = 0, then f is a polynomial mapping （see Ning et al. （2017））. In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan＇s theorem.

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .

Structure-Preserving Finite-Element Schemes for the Euler-Poisson Equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and selfgravitation modeling.The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law,as well as hyperbolic invariant domain properties,such as positivity of the density and a minimum principle of the specific entropy.A detailed discussion of algorithmic details is given,as well as proofs of the claimed properties.We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

Dynamics of a family of rational maps concerning renormalization transformation

Considering a family of rational maps Tnλconcerning renormalization transform ation,we give a perfect description about the dynamical properties of Tnλand the topological properties of the Fatou components F(Tnλ).Furthermore,we discuss the continuity of the Hausdorff dimension HD(J(Tnλ))about real param eter A.