New Asymptotic Results on Fermat-Wiles Theorem

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.

Small Modular Solutions to Fermat’s Last Theorem

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.

Whole Perfect Vectors and Fermat’s Last Theorem

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.

基于S-R和分解定理的二维几何非线性问题的虚单元法求解

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。

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

基于TDLAS的天然气泄漏范围重构方法

可调谐半导体激光光谱(TDLAS)技术因其灵出色的灵敏度、精度和抗干扰性的特性,现已被广泛应用于场站天然气泄漏的监测工作。为克服单台TDLAS激光遥测仪无法获得泄漏气体扩散影响范围的局限性,本文提出了一种基于多台TDLAS遥测仪协同的智能监测方法,通过在不同斜平面内对泄漏气团进行扫查,依据Brianchon定理获得不同斜平面内气体泄漏的外轮廓方程,重构天然气的三维空间泄漏影响区域,实时开展监测与预警。

四维Einstein-Gauss-Bonnet引力时空中非线性电磁场对光线偏折的影响

文章研究了含非线性电磁场的四维爱因斯坦-高斯-博内引力时空中的弱引力透镜现象。首先应用高斯-博内定理计算了含非线性电磁场的四维爱因斯坦-高斯-博内引力中的确切带电黑洞解得时空中光线的弱偏转角,并讨论了高斯-博内参数和非线性参数对偏转角的影响。此外,还利用类光测地线法计算了该黑洞的阴影半径。利用来自于事件视界望远镜的M87*黑洞观测数据限制了高斯-博内参数和磁荷,并讨论了非线性参数对观测值的影响。

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

N次幂下S-凸函数和其性质研究

以凸函数、S-凸函数、平方凸函数、平方S-凸函数的定义和性质为研究基础,根据从平方凸函数到平方S-凸函数的演变推进过程,采用了凸函数概念的代数不等式形式,给出了N次幂下S-凸函数的概念,并对其判别定理和性质进行了讨论,定义并证明了N次幂下S-凸函数的Jensen型不等式。研究表明,N次幂下S-凸函数包含了凸函数、平方凸函数、平方S-凸函数的性质,是凸函数的一种广义凸函数的推广,开辟了全新的推广途径。

THE GAUSS–GREEN THEOREM IN CLIFFORD ANALYSIS AND ITS APPLICATIONS

In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.

On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .

A Proof of Brouwer’s Fixed Point Theorem Using Sperner’s Lemma

This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.

Gravitation, Density Upper Limit and Quantization of Space

The singularity at distance r → 0 at the center of a spherically symmetric non-rotating, uncharged mass of radius R, is considered here. Under inverse square law force, the Schwarzschild metric, needs to be modified, to include Newton’s Shell Theorem (NST). By including NST for r, both Schwarzschild singularity at r = 2GM/c2 and at r → 0 singularities are removed from the metric. Near R → 0, the question of maximal density is considered based on Schwarzschild’s modified metric, and compared to the quantum limit of maximal mass density put by Planck’s quantum-based universal units. It is asserted, that General relativity, when combined with Planck’s universal units, inevitably leads to quantization of gravity.

具有饱和恢复率的SEIR时滞模型的行波解

研究一类具有饱和恢复率的SEIR时滞模型的行波解.首先,考虑一类二维系统初值问题的适定性;然后,通过构造一对有界的向量值上、下解得到一个闭凸集;最后,利用Schauder不动点定理证明:当基本再生数R^(0)>1,波速c>c^(*)时模型存在非平凡行波解.

数论中的Wilson定理通过群论中的Cauchy定理的证明

本文利用近世代数课程中有限群的Cauchy定理证明数论课程里的Wilson定理.

从第二类梯度算子和第二类积分定理到Gauss(球面)映射不变量

将第二类梯度算子、第二类积分定理、Gauss曲率相关的积分定理和Gauss(球面)映射相结合,证明了一系列Gauss(球面)映射不变量.从这些不变量中,得到一系列从原始曲面到(Gauss单位)球面的变换.这些不变量和变换,在几何学、物理学、生物力学和力学中,都有潜在的用途.

5维Gauss-Bonnet-AdS黑洞的热力学和临界现象

把宇宙学常数理解为压强,利用标准的热力学方法讨论了5维Gauss-Bonnet-Ad S黑洞系统的热力学和临界现象.把Bekenstein-Hawking熵作为黑洞系统的熵,发现在5维球对称的Gauss-Bonnet-Ad S黑洞中存在相变现象.进一步计算了相变点的临界指数,发现他们满足热力学普适标度规律.

一类广义Gauss型求积公式

基于被积函数在n次第一类和第二类Chebyshev多项式的零点处的差商,该本构造了两种Gauss型求积公式,这些求积公式包含了某些已知结果作为特例。更重要的是这些新结果与Gauss-Turán求积公式有密切的联系。

R2.1内给定平均曲率函数或Gauss曲率函数的一类特殊曲面的存在性定理

在R21内给出了具有给定平均曲率函数或Gauss曲率函数的绕零轴(0,t,t)(t∈(-∞,+∞))的光滑浸入曲面的存在性定理.

Gauss-Newton法的半局部收敛性

设f:Rn→Rm 是Frechet可微的 ,m≥n .则非线性最小二乘问题可描述为下面的极小化问题 :minF(x) :=12 f(x) Tf(x) .Gauss Newton法是求解非线性最小二乘问题的最基本的方法之一 ,其n + 1步迭代定义为 :xn + 1=xn - f′(xn) Tf′(x) -1f′(xn) Tf(xn) .本文主要研究解非线性最小二乘问题的Gauss Newton法的半局部收敛性 .假设f(x)在B(x0 ,r)内连续可导且f′(x0 )满秩 ,若f的导数满足Lipschitz连续F′(x) -f′(x′)≤γx -x′ , x ,x′∈B(x0 ,r) .在一个关于初始点x0 的判断准则c =f(x0 ) ,β =f′T(x0 )f′(x0 ) -1f′(x0 ) T ,β2 cγ <1 1 0下 ,Gauss Newton法产生的序列 {xn}收敛到一个驻点x ,从而给出了Gauss Newton法的半局部收敛性 .