This paper summarizes research intended to develop a pedagogically friendly argument that establishes the fact that (x,ex ) is never a rational point in the plane. A point (x, y)∈R2 is rational if both x and y are ra...This paper summarizes research intended to develop a pedagogically friendly argument that establishes the fact that (x,ex ) is never a rational point in the plane. A point (x, y)∈R2 is rational if both x and y are rational. Applying a method based on Hurwitz polynomials, the research establishes simple irrationality proofs for nonzero rational powers of e and logarithms of positive rationals (excluding one).展开更多
We compute rational points on real hyperelliptic curves of genus 3 defined on <img src="Edit_ff1a2758-8302-45a6-8c7e-a73bd35f12bd.png" width="20" height="18" alt="" /> who...We compute rational points on real hyperelliptic curves of genus 3 defined on <img src="Edit_ff1a2758-8302-45a6-8c7e-a73bd35f12bd.png" width="20" height="18" alt="" /> whose Jacobian have Mordell-Weil rank <em>r=0</em>. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and <span>the Chabauty-Coleman method to find <em>C </em>(<img src="Edit_243e29b4-1b26-469a-9e65-461ffac1e473.png" width="20" height="18" alt="" />)<span></span>. We run the algorithms in</span> Sage on 47 real hyperelliptic curves of genus 3.展开更多
This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of...This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.展开更多
In this paper we introduce a so called C-Matrix w.r.t a rational interpolation problem and study the relationship between the unattainable points and C-Matrix. Finally, we present a recursive algorithm on rational int...In this paper we introduce a so called C-Matrix w.r.t a rational interpolation problem and study the relationship between the unattainable points and C-Matrix. Finally, we present a recursive algorithm on rational interpolation.展开更多
Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exception...Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exceptional cases occur.展开更多
For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is...For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent re-展开更多
In this paper we study the number of rational points on some curves over finite fields. Moreover, zeta functions of the associated function fields are evaluated explicitly.
Many results on the arithmetic theory of elliptic curves have been obtained for elliptic curves with complex multiplication by Z[i], e.g. [1]—[3], which can be considered as the simplest case. The next simplest case ...Many results on the arithmetic theory of elliptic curves have been obtained for elliptic curves with complex multiplication by Z[i], e.g. [1]—[3], which can be considered as the simplest case. The next simplest case may be the elliptic curve with complex multiplication展开更多
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in...We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.展开更多
We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. W...We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.展开更多
This paper studies representation of rigid combination of a directed line and a reference point on it(here referred to as a "point-line") using dual quaternions.The geometric problem of rational ruled surfac...This paper studies representation of rigid combination of a directed line and a reference point on it(here referred to as a "point-line") using dual quaternions.The geometric problem of rational ruled surface design is viewed as the kinematic prob-lem of rational point-line motion design.By using the screw theory in kinematics,mappings from the spaces of lines and point-lines in Euclidean three-dimensional space into the hyperplanes in dual quaternion space are constructed,respectively.The problem of rational point-line motion design is then converted to that of projective Bézier or B-spline image curve design in hyperplane of dual quaternions.This kinematic method can unify the geometric design of ruled surfaces and tool path gen-eration for five-axis numerical control(NC) machining.展开更多
We classify,up to some lattice-theoretic equivalence,all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.
文摘This paper summarizes research intended to develop a pedagogically friendly argument that establishes the fact that (x,ex ) is never a rational point in the plane. A point (x, y)∈R2 is rational if both x and y are rational. Applying a method based on Hurwitz polynomials, the research establishes simple irrationality proofs for nonzero rational powers of e and logarithms of positive rationals (excluding one).
文摘We compute rational points on real hyperelliptic curves of genus 3 defined on <img src="Edit_ff1a2758-8302-45a6-8c7e-a73bd35f12bd.png" width="20" height="18" alt="" /> whose Jacobian have Mordell-Weil rank <em>r=0</em>. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and <span>the Chabauty-Coleman method to find <em>C </em>(<img src="Edit_243e29b4-1b26-469a-9e65-461ffac1e473.png" width="20" height="18" alt="" />)<span></span>. We run the algorithms in</span> Sage on 47 real hyperelliptic curves of genus 3.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11672257, 11772306, 11972173, and 12172340)the 5th 333 High-level Personnel Training Project of Jiangsu Province of China (Grant No. BRA2018324)。
文摘This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.
基金The NNSF (10471055) of China and the National Grand Fundamental Research 973 Program (2004CB318000) of China.
文摘In this paper we introduce a so called C-Matrix w.r.t a rational interpolation problem and study the relationship between the unattainable points and C-Matrix. Finally, we present a recursive algorithm on rational interpolation.
文摘Let R(z) be a rational function of degree d ≥ 2. Then R(z) has at least one repelling periodic point of given period k ≥ 2, unless k = 4 and d=2, or k= 3 and d ≤ 3, or k=2 and d≤8. Examples show that all exceptional cases occur.
文摘For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent re-
基金The authors would like to give thanks to the referees for many helpful suggestions. This work was jointly supported by the National Natural Science Foundation of China （11371208）, Zhejiang Provincial Natural Science Foundation of China （LY17A010008） and Ningbo Natural Science Foundation （2017A610134）, and sponsored by the K. C. Wong Magna Fund in Ningbo University.
基金supported by National Natural Science Foundation of China (Grant No.60903036)Natural Science Foundation of Jiangsu Province, China (Grant No. BK2009182)
文摘In this paper we study the number of rational points on some curves over finite fields. Moreover, zeta functions of the associated function fields are evaluated explicitly.
基金Project supported by the National Natural Science Foundation of China.
文摘Many results on the arithmetic theory of elliptic curves have been obtained for elliptic curves with complex multiplication by Z[i], e.g. [1]—[3], which can be considered as the simplest case. The next simplest case may be the elliptic curve with complex multiplication
文摘We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052)
文摘We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
基金supported by the National Natural Science Foundation of China(Grant Nos.50835004 and 51005087)the National Basic Research Program of China(Grant No.2011CB706804)
文摘This paper studies representation of rigid combination of a directed line and a reference point on it(here referred to as a "point-line") using dual quaternions.The geometric problem of rational ruled surface design is viewed as the kinematic prob-lem of rational point-line motion design.By using the screw theory in kinematics,mappings from the spaces of lines and point-lines in Euclidean three-dimensional space into the hyperplanes in dual quaternion space are constructed,respectively.The problem of rational point-line motion design is then converted to that of projective Bézier or B-spline image curve design in hyperplane of dual quaternions.This kinematic method can unify the geometric design of ruled surfaces and tool path gen-eration for five-axis numerical control(NC) machining.
文摘We classify,up to some lattice-theoretic equivalence,all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.
