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.

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.

参数化S谱幅值调制及轴承故障诊断应用

针对传统谱幅值调制方法易受噪声影响的问题,利用参数化S变换得到信号在时频域中的幅值,提出了一种参数化S谱幅值调制方法。该方法首先使用参数化S变换将信号转换到时频域并得到幅值和相位,然后将不同权重赋予时频域中的幅值以改变不同能量频率成分在信号中的占比,最后将调制后的幅值与原相位结合,使用参数化S逆变换重构一系列修正信号并计算其平方包络谱以提取故障特征。仿真和实验结果表明,该方法获得的幅值信息相比传统谱幅值调制方法更加准确和全面,对强噪声环境更具鲁棒性,能够有效实现滚动轴承的外圈、内圈和复合故障诊断。将所提方法与传统谱幅值调制方法和快速谱峭度方法进行对比,证明了参数化S谱幅值调制既能检测强噪声环境下的轴承故障信息,又能同时提取多种故障分量,在滚动轴承的故障特征提取中更具优越性。

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.

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.

Investigation of Probability Generating Function in an Interdependent <i>M/M/</i>1:(∞;GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. In this paper we apply the Rothe’s Fixed Point Theorem to prove the interior approximate controllability of the following Benjamin Bona-Mohany(BBM) type equation with impulses and delay where and are constants, Ω is a domain in , ω is an open non-empty subset of Ω , denotes the characteristic function of the set ω , the distributed control , are continuous functions and the nonlinear functions are smooth enough functions satisfying some additional conditions.

S模式二次雷达询问策略的研究和改进

通常常规二次雷达(SSR)工作在A/C模式,S模式二次雷达工作在交互模式,通过不同的询问策略获得应答信号。在实际应用中发现,特殊应答机对于交互模式的应答存在丢点问题。本文通过对现有S模式二次雷达询问策略的分析和研究,在国际民航组织二次雷达规范的基础上提出了改进的询问策略,解决了S模式二次雷达与传统A/C模式二次雷达兼容过程中偶尔出现的目标丢失问题,为S模式二次雷达的运用提供了建设性意见。

关于罗尔(Rolle)中值定理条件的研究

罗尔中值定理指出,当函数f(x)满足三个特定条件时,在区间内部至少存在一点ξ,使得f′(ξ)=0,本文针对在区间[a,b]端点处不连续的函数以及无穷区间上的可导函数的相关问题作了进一步研究,所得结论推广和完善了文献中相应的定理。

Rolle定理的推广和Rota问题

本文将经典的Rolle定理推广到一类微分算子，并对哑演算中Rota提出的至今尚未解决的广义Taylor展开式的余项估计问题所派生的问题作了研究．

Rolle中值定理的一个新证明

本文给出了Rolle中值定理的一种新的证明方法

多项式插值与Rolle定理

本文利用多项式插值 ,对微分学和积分学中的一些典型题目 ,给出了统一的处理方法 .这种方法易于模仿 ,有一定的适用性 .

Rolle定理的推广

本文对一元函数的可微分性的Rolle中值定理进行了推广。

Rolle定理证明及应用的探讨

对Rolle定理的两种不同证明方法,即闭区间套定理、最大值与最小值定理证明进行了深入的探讨,并对定理的推广及应用作了进一步的分析研究。

关于Rolle定理的评注

Rolle定理是微分中值的基本定理 ,在微积分学中起着重要作用。关于 Rolle定理有着多种形式的推广。本文得到了如下有意义的结论。设函数 y=f( x)在区间 I的内部可导 ,在区间 I上连续 ,若在区间 I上存在三个点 x1 ,x2 ,x3 ,x1 <x2 <x3 ,使f( x2 )≥ max{ f( x1 ) ,f( x3 ) }或者 f( x2 )≤ min{ f( x1 ) ,f( x3 ) } ,则在区间 I内至少存在一点ξ,使 f (ξ) =0。此结论在证明关于“ξ等式”时十分方便 ,简洁。不论在理论上 。