The Nother and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nother type theorems for Cμpiecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible Cμpiecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cμpiecewise algebraic curve is established.     

Numerical realization of the conditions of Max NSther＇s residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nother＇s residual intersection theorem. The numerical realization relies on obtaining the inter- section of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determin- ing the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

Numerical realization of the conditions of Max Nther's residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nther's residual intersection theorem. The numerical realization relies on obtaining the intersection of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determining the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic,even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

A kinematic formula for integral invariant of degree 4 in real space form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.     

Nther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space.In this paper,using the properties of bivariate splines,the Nther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bézier curve, usually related to an explicit function,is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized monotonicity, called direction monotonicity, is introduced for rational Bézier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine coordinate systems, and it includes the traditional monotonicity as a subcase. By means of it,proper affine coordinate systems may be chosen to make some rational Bézier curves monotonic.Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.     

The local classifications of the ruled surfaces of normals and binormals of a regular space curve

The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.     

Morgan-Scott型剖分上的样条空间的奇异性

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.     

实分片代数曲线的拓扑结构

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.By analyzing the topology of real algebraic curves on the triangles,a practi-caUy algrithm for analyzing the topology of piecewise algebraic curves is given.The algrithm produces a planar graph which is topologically equivalent to the piecewise algebraic curve.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

ZONAL SPHERICAL POLYNOMIALS WITH MINIMAL L_1-NORM

Radial functions have become a useful tool in numerical mathematics. On the sphere they have to be identified with the zonal functions. We investigate zonal polynomials with mass concentration at the pole, in the sense of their L1-norm is attaining the minimum value. Such polynomials satisfy a complicated system of nonlinear e-quations (algebraic if the space dimension is odd, only) and also a singular differential equation of third order. The exact order of decay of the minimum value with respect to the polynomial degree is determined. By our results we can prove that some nodal systems on the sphere, which are defined by a minimum-property, are providing fundamental matrices which are diagonal-dominant or bounded with respect to the ∞-norm, at least, as the polynomial degree tends to infinity.     

Approximation of conic section by quartic Bzier curve with endpoints continuity condition

A new method for approximation of conic section by quartic B′ezier curve is presented, based on the quartic B′ezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme,and the effectiveness of this method is demonstrated by some numerical examples.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

An Extension of the Turan Inequality in L_p-space for 0

周颂平 《数学研究与评论》1986,(2)

Pythagorean Bézier速端曲线及其等距线

The Pythagorean Bézier hodograph curves are Bézier curves {x(t), y(t)}, whose hodograph (first-order parametric derivative) components satisfy the Phythagorean condition x'2(t) +y'2(t) = σ'(t) for some polynomial σ(t). For nth degree PB curve (n is odd), its offset curve is represented by rational Bézier curve with (2n - 1)th degree and arc length by polynomial. Specially, the properties of cubic PB curve are studied, its geometric features are discussed and its quasi-Hermite interpolating curve and GC1 composite cubic PB curve are also constructed.     

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.     

A new type of the generalized Bézier curves

In this paper, we improve the generalized Bernstein basis functions introduced by Han, et al. The new basis functions not only inherit the most properties of the classical Bernstein basis functions, but also reserve the shape parameters that are similar to the shape parameters of the generalized Bernstein basis functions. The degree elevation algorithm and the conversion formulae between the new basis functions and the classical Bernstein basis functions are obtained. Also the new Q-Bézier curve and surface...     

Hilbert投影距离与Banach空间的范数

In this paper, we establish some relations between the Hilbert＇s projective metric and the norm on a Banach space and show that the metric and the norm induce equivalent convergences at certain set. As applications, we utilize the main results to discuss the eigenvalue problems for a class of positive homogeneous operators of degree a and the positive solutions for a class of nonlinear algebraic system.     

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q ∞.