We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Dio...We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.展开更多
The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infini...The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.展开更多
A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm de...A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.展开更多
Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of ...Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#'_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.展开更多
We prove a Maschke type theorem for Doi-Hopf π-modules. A sufficient condition for having a Maschke type property is that there exists a suitable total integral map for the Doi-Hopf π-modules in question. The applic...We prove a Maschke type theorem for Doi-Hopf π-modules. A sufficient condition for having a Maschke type property is that there exists a suitable total integral map for the Doi-Hopf π-modules in question. The applications of the results are considered. Finally, As an application of the existence of total integral, we prove that α∈π Ca A is a generator in the category π-Cu(H)A.展开更多
In this paper, we mainly generalize a Maschke type theorem to the setting of a weak Hopf group coalgebra. First we introduce the notion of a weak Hopf group coalgebra as a generalization of Hopf group coalgebra introd...In this paper, we mainly generalize a Maschke type theorem to the setting of a weak Hopf group coalgebra. First we introduce the notion of a weak Hopf group coalgebra as a generalization of Hopf group coalgebra introduced in [7] and a weak Hopf algebra introduced in [2]. And we study some basic properties of weak Hopf group coalgebras. Next we aim at finding some sucient conditions under which an epimorphism of weak (H, A) Hopf π-comodule splits if it splits as an A-module morphism and give an application of our results.展开更多
应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝...应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。展开更多
文摘We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.
文摘The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.
文摘A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.
基金The National Natural Science Foundation of China(No.11371088)the Natural Science Foundation of Jiangsu Province(No.BK2012736)+1 种基金the Fundamental Research Funds for the Central Universitiesthe Research Innovation Program for College Graduates of Jiangsu Province(No.KYLX15_0109)
文摘Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#'_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.
文摘We prove a Maschke type theorem for Doi-Hopf π-modules. A sufficient condition for having a Maschke type property is that there exists a suitable total integral map for the Doi-Hopf π-modules in question. The applications of the results are considered. Finally, As an application of the existence of total integral, we prove that α∈π Ca A is a generator in the category π-Cu(H)A.
基金Supported by the Natural Science Foundation of Shandong Province(ZR2012AL02)
文摘In this paper, we mainly generalize a Maschke type theorem to the setting of a weak Hopf group coalgebra. First we introduce the notion of a weak Hopf group coalgebra as a generalization of Hopf group coalgebra introduced in [7] and a weak Hopf algebra introduced in [2]. And we study some basic properties of weak Hopf group coalgebras. Next we aim at finding some sucient conditions under which an epimorphism of weak (H, A) Hopf π-comodule splits if it splits as an A-module morphism and give an application of our results.
文摘应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。