Classification of signature-only signature models

We introduce a set of criterions for classifying signature-only signature models. By the criterions, we classify signature models into 5 basic types and 69 general classes. Theoretically, 21141 kinds of signature models can be derived by appropriately combining different general classes. The result comprises almost existing signature models. It will be helpful for exploring new signature models. To the best of our knowledge, it is the first time for investigation of the problem of classification of signature-only signature models.

On the Topology and Isotopic Meshing of Plane Algebraic Curves

This paper presents a symbolic algorithm to compute the topology of a plane curve.This is a full version of the authors’CASC15 paper.The algorithm mainly involves resultant computations and real root isolation for univariate polynomials.Compared to other symbolic methods based on elimination techniques,the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system f(α,y),?f/?y(α,y)and simultaneously obtain numerous simple roots of f(α,y)=0 on theαfiber.This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α,y)=0.After the topology is computed,a revised Newton’s method is presented to compute an isotopic meshing of the plane algebraic curve.Though the approximation method is numerical,the authors can ensure that the proposed method is a certified one,and the meshing is topologically correct.Several nontrivial examples confirm that the proposed algorithm performs well.

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).

Isotopic Meshing of a Real Algebraic Space Curve

This paper presents a new algorithm for computing the topology of an algebraic space curve.Based on an efficient weak generic position-checking method and a method for solving bivariate polynomial systems,the authors give a first deterministic and efficient algorithm to compute the topology of an algebraic space curve.Compared to extant methods,the new algorithm is efficient for two reasons.The bit size of the coefficients appearing in the sheared polynomials are greatly improved.The other is that one projection is enough for most general cases in the new algorithm.After the topology of an algebraic space curve is given,the authors also provide an isotopic-meshing(approximation)of the space curve.Moreover,an approximation of the algebraic space curve can be generated automatically if the approximations of two projected plane curves are first computed.This is also an advantage of our method.Many non-trivial experiments show the efficiency of the algorithm.

Mechanical theorem proving in the surfaces using the characteristic set method and Wronskian determinant

In this paper, we generalize the method of mechanical theorem proving in curves to prove theorems about surfaces in differential geometry with a mechanical procedure. We improve the classical result on Wronskian determinant, which can be used to decide whether the elements in a partial differential field are linearly dependent over its constant field. Based on Wronskian determinant, we can describe the geometry statements in the surfaces by an algebraic language and then prove them by the characteristic set method.