Hyperfinite Interpolation,Wu's Method and Blending of Implicit Algebraic Surfaces

one of the central questions in CAGD[1] is blending surfaces whichprovides the theoretical baJsis for the design technology of space surfaces. We willdiscuss the general theories and algorithms for multivariate hyperfinite interpolationand their aPplication to the blending of implicit algebraic surfaces, and investigate theexistence conditions of hyperfinite interpolation. Based on Wu's theory on blendingimplicit algebraic surfaces, the problem of blending two quadric surfaces is studied.The conditions for the coefficient of gi under which there exists the cubic blendingsurface S(f) (the lowest degree) are obtained and the concrete expressions of f arepresented if they exist. These results can be applied directly to CAGD.

ON EFFECTIVE NON-VANISHING OF WEIL DIVISORS ON ALGEBRAIC SURFACES

The author gives a counterexample and some conclusions for effective non-vanishing of Weil divisors on algebraic surfaces.

INVARIANT ALGEBRAIC SURFACES OF SOME DYNAMICAL SYSTEM

In this paper, we study the invariant algebraic surfaces of a system, which generalizes the Lorenz system. Using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations, we characterize all the Darboux invariants, the irreducible Darboux polynomials, the rational first integrals and the algebraic integrability of this system.

DEGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE

We give a local analytic characterization that a minimal surface in the 3-sphere S3 C R4 defined by an irreducible cubic polynomial is one of the Lawson＇s minimal tori. This provides an alternative proof of the result by Perdomo （Characterization of order 3 algebraic immersed minimal surfaces of S3, Geom. Dedicata 129 （2007）, 23 34）.

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis （Adv. Comput. Math., 17 （2002）, pp. 135-152） based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Juttler＇s method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.

On some examples of obstructed irregular surfaces

We determine the base space of the Kuranishi family of some complete intersections in the product of an abelian variety and a projective space.As a consequence,we obtain new examples of obstructed irregular surfaces with ample canonical bundle and maximal Albanese dimension.

Two-dimensional Irreducible Algebraic Semigroups

We study two-dimensional irreducible projective smooth algebraic semigroups. Minimal surface semigroups with Kodaira dimension at most one are partially classified. We also calculate the local dimension of the moduli scheme parameterizing all algebraic semigroup laws on a fixed minimal surface.

Proper Reparametrization of Rational Ruled Surface

In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.

On the Darboux Integrability of the Hindmarsh–Rose Burster

We study the Hindmarsh-Rose burster which can be described by the differential system x^·=y-x^3＋bx^2＋I-z,y^·=1-5x^2-y,z^·=μ（s（x-x0）-z）where b, I, μ, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.