Cubic Algebraic Spline Curves Design

In this paper we propose a construction method of the planar cubic algebraic splinecurve with endpoint interpolation conditions and a specific analysis of its properties. Thepiecewise cubic algebraic curve has G2 continuous contact with the control polygon at twoendpoints and is G2 continuous between each segments of itself. The process of this method issimple and clear, and provides a new way of thinking to design implicit curves.

Quadratic Algebraic Curve Approximate Implicitization for Planar Parametric Curves

In this paper, a type of preserving GC1 quadratic algebraic polynomial curve approximate implicitization method for parametric curves is presented The coefficients of the implicit polynomial are determined by the GC1 continuity conditions and an optimal function's minimization Numerical examples show that this method is effective

Optimal parameterization of conic curves

Optimal parameterization of specified segment on the algebraic curves is a hot issue in CAGD and CG. Take the optimal approximation of arc-length parameterization as the criterion of optimal parameterization, and the optimal or close to optimal rational parameterization formula of any specified segment on the conic curves is obtained. The new method proposed in this paper has ad- vantage in quantity of calculation and has strong self-adaptability. Finally, a experimental comparison of the results obtained by this method and by the traditional parametric algorithm is conducted.

Optimal Rational Parameterization of Quadratic Curves Based on the Geographic Information of Both Ends

In this paper, we rewrote the equation of algebraic curve segmentswith the geometric informationonboth ends. The optimal or nearly optimal rationalparametric equation is determinedbythe principle that parametricspeedsat both endsareequal. Comparing withotherliteratures, the methodofthis paper has advantage in efficiency andiseasy to realize. The equation of optimal rational parameterization can be obtained directly by the information of both ends. Large numbers ofexperimental data show that our method hasbeen given withmore self-adaptability and accuracy than that ofotherliteratures, and if the parametricspeedat any end reaches its maximum or minimum value, the parameterization is optimal; otherwise itis close tooptimal rational parameterization.

PIECEWISE RATIONAL APPROXIMATIONS OF REAL ALGEBRAIC CURVES

We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C-1 continuity between pieces. The simpler cases of C-1 and C-0 continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.

LIMIT CYCLES AND INVARIANT PARABOLA IN A KUKLES SYSTEM OF DEGREE THREE

In this article,the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola.For this class of second-order differential systems,it is shown that for certain values of the parameters the invariant parabola coexists with a center.For other values it can coexist with one,two or three small amplitude limit cycles which are constructed by Hopf bifurcation.This result gives an answer for the question given in[4],about the existence of limit cycles for such class of system.

Nother-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nther-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 Nther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.

INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIALDIFFERENTIAL SYSTEMS

We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.

Hilbert's Sixteenth Problem and Its Generalization

This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree n are discussed. The maximum number of sheets for an algebraic equation of degree n and the maximum number of limit cycles for a planar algebraic autonomous system are achieved. The author also considers different generalizations and some related problems.

The Cayley-Bacharach Theorem for Continuous Piecewise Algebraic Curves over Cross-cut Triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.

Exact parametric representations of orbits defined by cubic Hamiltonian

In this paper,we show that for any given planar cubic algebraic curves defined by a quadratic Hamiltonian vector field,we can always have their exact explicit parametric representations. We use a model of micro-structured solid to show an application of our conclusions.

Packing Superballs from Codes and Algebraic Curves

In the present paper, we make use of codes with good parameters and algebraic curves over finite fields with many rational points to construct dense packings of superballs. It turns out that our packing density is quite reasonable. In particular, we improve some values for the best-known lower bounds on packing density.

Numerical realization of the conditions of Max Nther'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.

The Bezout Number of Piecewise Algebraic Curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.

Some Results on Special Stable Vector Bundles of Rank 3 on Algebraic Curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.

ON THE ISOMORPHISMS OF SMOOTH ALGEBRAIC CURVES(Ⅰ)

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On the Complexity of Computing the Topology of Real Algebraic Space Curves

This paper presents an algorithm to compute the topology of an algebraic space curve.This is a modified version of the previous algorithm.Furthermore,the authors also analyse the bit complexity of the algorithm,which is O(N^(20)),where N=max{d,τ},d andτare the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve.To our knowledge,this is the best bound among the existing work.It gains the existing results at least N^(2).Meanwhile,the paper contains some contents of the conference papers(CASC 2014 and SNC 2014).

General Solutions of First-Order Algebraic ODEs in Simple Constant Extensions

If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this field.Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial.As for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs)of order one,and studies their Liouvillian general solutions.In particular,let F(y,w)=0 be a rational algebraic curve over C.The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y,y′)=0 to have a Liouvillian solution over C.Moreover,the authors show that a Liouvillian solutionαof this equation is either an algebraic function over C(x)or an algebraic function over C(exp(ax)).As a byproduct,these results lead to an algorithm for determining a Liouvillian general solution of an autonomous AODE of order one of genus zero.Rational parametrizations of rational algebraic curves play an important role on this method.