Isolated complex singularities and their CR links

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Further advances on low-energy lunar impact dynamics

We extend the analysis, started in a previous work [1], concerning the formation of lunar impact craters due to low-energy trajectories. First, we adopt the Circular Restricted Three-Body Problem and consider different choices of initial conditions inside the stable invariant manifold associated with the central invariant one in the neighborhood of the L₂ equilibrium point in the Earth-Moon system. Then we move to the Bicircular Restricted Four-Body Problem to study the effect of the Sun on the distribution of impacts on the Moon’s surface.     

New exact solutions of (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation using a combination of lie symmetry and singular manifold methods

A new combination of Lie symmetry and Singular Manifold methods has been employed to study (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (KP). Infinite-dimensional space of Lie vectors has been established. Single and dual linear combinations of Lie vectors are used after appropriate calculations of the arbitrary functions to reduce the equation to an ordinary differential equation (ODE). The resulting ODE is then analytically solved through the singular manifold method which resulted in a Bäcklund truncated series with seminal analysis leading to a Schwarzian differential equation in the Eigenfunction φ (η). Solving this differential equation leads to new analytical solutions.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

Manifolds with Boundary and of Bounded Geometry

For non–compact manifolds with boundary we prove that bounded geometry defined by coordinate–free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry and we study the change of geodesic coordinate maps.     

Geometrical methods for analyzing the optimal management of tipping point dynamics

Natural resources are not infinitely resilient and should not be modeled as being such. Finitely resilient resources feature tipping points and history dependence. This paper provides a didactical discussion of mathematical methods that are needed to understand the optimal management of such resources: viscosity solutions of Hamilton–Jacobi–Bellman equations, the costate equation and the associated canonical equations, exact root counting, and geometrical methods to analyze the geometry of the invariant manifolds of the canonical equations. Recommendations for Resource Managers

• Management of natural resources has to take into account the possible breakdown of resilience and induced regime shifts.
• Depending on the characteristics of the resource and on its present and future economic importance, either for all initial states the same kind of management policy is optimal, or the type of the optimal management policy depends on the initial state.
• Modeling should reflect the finiteness of the data.

     

All flat three-manifolds appear as cusps of hyperbolic four-manifolds

There are ten diffeomorphism classes of compact, flat 3-manifolds. It has been conjectured that each of these occurs as the boundary of a 4-manifold whose interior admits a complete, hyperbolic structure of finite volume. This paper provides evidence in support of the conjecture. In particular, each diffeomorphism class of compact, flat 3-manifolds is shown to appear as one of the cusps of a complete, finite-volume, hyperbolic 4-manifold. This is done with a construction that uses special coverings of ³ by 3-balls. A further consequence of the construction is a finer result about the geometric structures which can be induced on cusps of complete, finite-volume, hyperbolic 4-manifolds. Using Mostow's Rigidity Theorem, one can show that not every flat structure occurs in this way. However, the fact that the flat structures induced on cusps of such 4-manifolds are dense in their respective moduli spaces follows from the construction.     

紧致对称空间的热核、特征值和特征函数

本文利用群表示论研究李群以及紧致对称空间的热核，特征值与特征函数．特别讨论了复格拉斯曼流形以及流形Sp（n）／U（n）上特征值及特征函数。     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

On the elliptic equation on complete Riemannian manifolds and their geometric applications

We study the elliptic equation on complete noncompact Riemannian manifolds with nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space and the hyperbolic space are carried out when is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.