Rational Points on Genus 3 Real Hyperelliptic Curves

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

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.

Rational double points on Enriques surfaces

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.

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.

《高等职业教育专科英语课程标准(2021年版)》课程结构的理据和要点

高职院校贯彻《高等职业教育专科英语课程标准(2021年版)》时,面临的首要任务可能就是根据新课标的课程结构要求改造和完善校本英语课程体系。新课标课程结构的设置建立在职业类型定位、素养方向引领、多元需求满足和客观条件兼顾等理据之上,内涵要点包括高职英语的必要性、学时安排、课程类型和课程体系建设。课程结构应和课程内容、课程目标、学业质量等协同落实,才能最大限度发挥新课标的作用。

The Number of Solutions of Certain Equations over Finite Fields

Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q elements.Let N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1)x_(2)…x_(n)x^(k_(n+1))_(n+1)…x^(k_(t))_(t)over the finite field F_(q),with n≥2,t>n,k,and k_(j)(n+1≤j≤t),m_(i)(1≤i≤n)are positive integers.In this paper,we find formulas for N_(q)when there is a positive integer l such that dD|(p^(l)+1),where D=1 cm[d_(1),…,d_(n)],d=gcd(n∑i=1M/m_(i)-kM,(q-1)/D),M=1 cm[m_(1),…,m_(n)],d_(j)=gcd(m_(j),q-1),1≤j≤n.And we determine N_(q)explicitly under certain cases.This extends Markoff-Hurwitz-type equations over finite field.

Research on a Class of Equations overFinite Fields

Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of Fq.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over Fq：f（x 1,x 2,...,x n）=（a1 x1 x2d＋a2 x2 x3d...＋a（n-1）x（n-1）xnd＋an xn x1d）λ-bx1（d1）x2d2...xn（dn）,with ai,b∈Fq＊,n≥2,λ〉0 being positive integers,and d,di being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and h1,h2∈Fq[x 1,x 2,...,x n].It extends the results of the Markoff-Hurwitz-type equations.

针灸临床辨证论治的思考(英文)

当今针灸临床辨证论治的现状已引起国内外众多学者的担忧,日本国针灸同行指出临床中存在的诸多弊端应引起同行业人士足够的重视。应日方的几点提问,提出了当前存在的主要问题并分析其中仁者见仁、智者见智的某些看法。所谓"辨证论治"是在临床"理、法、方、穴、术"科学有序的具体运用中得以体现的。但要达到有效提高针灸疗效的目的,需在临床技能、作用规律、适应病证、影响因素等诸方面多下功夫。