Horizontal Isometric Maps and Gauss Maps

In 1970,Rub and Vilms [1]proved a famous result that the Gauss map ofan lsometric immersion into an Euclidean space is harmonic if and only if theimmersion has parallel mean curvature.Here we generalized the resutlt from“iso-     

The Enumeration of General Rooted Planar Maps

This paper provides some functional equations satisfied by the generating functions for enumerating general rooted planar maps with up to three parameters. Furthermore, the generating functions can be obtained explicitly by employing the Lagrangian inversion. This is also an answer to an open problem in 1989.     

Singular surfaces and the rigidity of ℙ3

In this paper we give two elementary proofs for the rigidity of the complex projective space ₃ with respect to global deformations.     

Structure properties of laminar currents on ℙ2

We study the structure of a class of laminar closed positive currents on ℂℙ², naturally appearing in birational dynamics. We prove such a current admits natural non intersecting leaves, that are closed under analytic continuation. As a consequence it can be seen as a foliation cycle a weak lamination.     

Standard Classes on the Blow-Up of ℙ n at Points in Very General Position

We study (?1)-classes on the blow-up X of ? ⁿ at points in very general position; in the case n = 2, we give a new proof of the equivalence of two conjectures about the dimension of the class of a divisor on X, and we prove that h ¹(D) = 0 for any nef D which is a nonnegative sum of (?1)-curves. For n = 3, we fill a gap in the formulation of a conjecture about the dimension of a class. We provide algorithms and Maple code based on these conjectures.     

Plane curves whose singular points are cusps and triple coverings of ℙ2

We study a plane curve C whose singular points are cusps and the surface which is a triple covering of ² branched along C. As a result, especially we obtain an inequality for the sum of the Milnor numbers at the singularities of C and new surfaces of general type.     

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes. Work supported by DGICYT Grant BFM2003–02037/MATE.     

Lower degree curves in X3,3 ⊂ ℙ2 × ℙ2

Science China Mathematics - In this paper we study low-degree low-genus curves in a generic hypersurface X of degree (3, 3) in ?2 × ?2. We prove that the genus 0 and genus 1 curves...     

Postulation of disjoint unions of planes and lines in ℙ5

Here we study the postulation of general disjoint unions of planes and lines in ?⁵ using the Horace Method and some unreduced structures called here 2-sundials.     

Flat deformations of ℙ n

In this paper we study projective flat deformations of ? ⁿ . We prove that the singular fibers of a projective flat deformation of ? ⁿ appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of ? ⁿ with smooth total space, and discuss flatness criteria.     

Attracting basins in ℙ2

We prove the existence of infinitely many attracting basins for some holomorphic mappings in ℙ². We also show that if a family of mappings has a complex generic homoclinic tangency, then some of the mappings in the family have an attractive periodic fixed point.     

A Remark on Kleiman–Piene's Question for Gauss Maps

We discuss some algebraic properties of the monoid generated by (left) translations in left distributive structures.This furnishes methods for enriching the original structure with a compatible associative product.     

Smooth non-special surfaces of ℙ4

We consider smooth non-degenerate surfaces in ℙ⁴, and prove that there is a finite number of such surfaces which are:

Surjectivity of Generaized Expanding Maps and Some Characterizations of Monotone Operators

In [1] L. Nirenberg stated the following problem: Suppose T is a continuous map H→H (H is a Hilbert space), which is expanding,i.e. and T(0)=0.Suppose T map a neighborhood of the origin onto a neighborhood of the origin. Does T maps H onto H? In [2] Kung-Ching Chang and Li Shujie answered positively the problem when T is differentiable.     

Problems on Scales and Maps

Imaging that there is a part of the map of Sentosa Island.It reasonably represents the rela- tive positions of interest.The scale shown at the bottom centre of this map gives a measure of the ratio of the distance on the map to the actural distance. In this case,4 cm on the map represents 1 km.since a scale can be expressed in a ratio form, we have:     

Self Maps and Margolis Killing Constructionin the Category of Spectra

ＳｅｌｆＭａｐｓａｎｄＭａｒｇｏｌｉｓＫｉｌｌｉｎｇＣｏｎｓｔｒｕｃｔｉｏｎｉｎｔｈｅ　Ｃａｔｅｇｏｒｙ　ｏｆ　ＳｐｅｃｔｒａＬｉｎＪｉｎｋｕｎ（林金坤）ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｎｋａｉＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ...     

Moment Maps and Equivariant Volumes

The study of the volume of big line bundles on a complex projective manifold M has been one of the main veins in the recent interest in the asymptotic properties of linear series. In this article, we consider an equivariant version of this problem, in the presence of a linear action of a reductive group on M.     

Generalized Gauss–Radau and Gauss–Lobatto Formulae

Computational methods are developed for generating Gauss-type quadrature formulae having nodes of arbitrary multiplicity at one or both end points of the interval of integration. Positivity properties of the boundary weights are investigated numerically, and related conjectures are formulated. Applications are made to moment-preserving spline approximation. AMS subject classification (2000) 65D32, 41A15.