Geometry of plane curves

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.     

Geometry of obstructed families of curves

We study certain equisingular families of curves with quasihomogeneous singularity of minimal obstructness (i.e. h¹=1). We show that our families always have expected codimension. Moreover they are either non-reduced with smooth reduction or decompose into two smooth components of expected codimension that intersect non-transversally or are reduced irreducible non-smooth varieties which have smooth singular locus with sectional singularity of type A₁. On the other hand there is an example of an equisingular family of curves with multiple quasihomogeneous singularities of minimal obstructness which is smooth but has wrong codimension. We use algorithms of computer algebra as a technical tool.     

六次PH曲线C~1 Hermite插值

本文讨论六次PH(pythagorean hodograph)曲线的Hermite插值问题.六次PH曲线可以分为两种类型,本文使用参数曲线的复数表示形式,分别给出这两类曲线的构造方法.在给定C1连续的Hermite条件下,需要指定一个自由参数以确定插值曲线,本文进一步阐述这个自由参数的几何意义.由于六次PH曲线是非正则曲线,对于第一类曲线,不易控制奇异点在曲线中的位置;而对于第二类曲线,奇异点可以在构造过程中显式地被指定,因此可以有效地避免其在特定曲线段上的出现.     

Geometry of curves in generalized flag varieties

The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.     

Algebraic characterization of generic strongly semi-regular rational PH plane curves

In this paper, we introduce a new algebraic method to characterize rational PH plane curves. And using this method, we study the algebraic characterization of generic strongly regular rational plane PH curves expressed in the complex formalism which is introduced by R. T. Farouki. We prove that generic strongly semi-regular rational PH plane curves are completely characterized by solving a simple functional equation ${\mathcal{H}}(f,g) = h^2 $ whereh is a complex polynomial and ${\mathcal{H}}$ is a bi-linear operator defined by ${\mathcal{H}}(f,g) = f'g - fg'$ for complex polynomialsf, g.     

Geometry and algebra of real forms of complex curves

Let Y be a complex algebraic curve and let be the set of all real algebraic curves with complexification , such that the real points divide . We find all such families [Y]. According to Harnak theorem a number of connected components of satisfies by the inequality , where g is the genus of Y. We prove that and these estimates are exact. Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Geometry of tubes about characteristic curves on Sasakian manifolds

One studies, using Riemannian foliation theory, some aspects of the intrinsic and extrinsic geometry of small tubes about the flow lines of the characteristic vector field on a Sasakian manifold. In particular, one focuses on some characteristic properties of the shape operator and the Ricci operator of these tubes for the classes of ?-symmetric spaces and Sasakian space forms.     

Geometry of the graphs of nonseparating curves: covers and boundaries

Geometriae Dedicata - We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite...     

Geometry of families of nodal curves on the blown-up projective plane

Let be the projective plane blown up at generic points. Denote by the strict transform of a generic straight line on and the exceptional divisors of the blown-up points on respectively. We consider the variety of all irreducible curves in with nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For we give the complete answer concerning the existence of nodal curves in .

     

Wavelet feature parameters representations of open planar curves

The use of Haar wavelets to describe open planar curves is presented in this paper. The geometric significance of a wavelet approximation and a wavelet detail for open curves in the complex plane are given. By analysing the internal relationship of the wavelet details, a normalised process is established for extracting the wavelet feature parameters of open curves. The application of this method to non-periodic design requirements of the linkage mechanism dimensional synthesis problem is presented. Two examples are proposed to demonstrate the feasibility of this method.     

Geometry of curves with exceptional secant planes: linear series along the general curve

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill?CNoether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d ? 2)-planes to (2d ? 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.     

A class of spline curves with four local shape parameters

A class of spline curves with four local shape parameters, which includes the quartic spline curves with three local shape parameters given in Han [Xuli Han. A class of general quartic spline curves with shape parameters. Comput. Aided Geom. Design, 28: 151–163 (2011)], is proposed. Without solving a linear system, the spline curves can be used to interpolate sets of points with C² continuity partly or entirely. The shape parameters have a predictable adjusting role on the spline curves.     

Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2,5)

We prove that the moduli spaces of rational curves of degree at most 3 in linear sections of the Grassmannian $\mathrm{G}\mathrm{r}(2,5)$ are all rational varieties. We also study their compactifications and birational geometry.     

Geometry of Hugoniot curves in 2 x 2 systems of hyperbolic conservation laws with quadratic flux functions

^** Email: asakura{at}isc.osakac.ac.jp^*** Email: yamazaki{at}math.tsukuba.ac.jp This note analyzes a simple discontinuous solution to hyperbolic2 x 2 systems of conservation laws having quadratic flux functionswith an isolated umbilic point where the characteristic speedsare equal. We study the Hugoniot curves in Schaeffer & Shearer'scase I and II which are relevant to the three-phase Buckley–Leverettmodel for oil reservoir flow. The compressive and overcompressiveparts are determined. The wave curves through the umbilic pointare discussed and their compressive and overcompressive partsare also determined.     

The relation between rigid-analytic and algebraic deformation parameters for Artin-Schreier-Mumford curves

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.     

Steady-state probabilities of the PH/PH/1 queue

It is proven that the steady-state probability of the PH/PH/1 queue is a linear combination of product forms. The method of linear combination of product forms is introduced, and simple formulae are obtained.     

离散时间服务台可修的排队系统MAP／PH（PH／PH）／1

本文研究离散时间可修排队系统,其中顾客的输入过程为离散马尔可夫到达过程（MAP）,服务台的寿命,服务台的顾客的服务时间和修理时间均为离散位相型（PH）变量,首先我们考虑广义服务过程,证明它是离散MAP,然后运用阵阵几何解理论,我们给出了系统的稳态队长分布和稳态等待时间分布,同时给出了系统的稳态可用度这一可靠性指标。