On the Weyl’s Lemma for Triharmonic Functions

In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.

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.

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.

Identities on q-Harmonic Numbers

With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.

S-引理及其进展

S-引理是运筹学领域和控制论领域中一个十分重要的定理。本文首先从一个非线性控制系统的全局渐近稳定性分析出发,引出S-过程和S-引理,以及两者之间的联系与区别。接着我们介绍了S-引理的基本内容和最新进展,以及它在复数域和四元数集合上的推广。最后,我们介绍了将齐次型S-引理推广到任意多个对称(或厄米特)矩阵时的一些基本结果。

S形空间曲线顶管开挖引起的地表沉降计算——以宁波市潘桥变电站迁建工程为例

为研究S形空间曲线顶管隧道施工引起的地表沉降规律,基于Mindlin弹性力学解和随机介质理论,构建S形空间曲线顶管施工引起的地表沉降计算模型,获得顶管施工各参数和空间曲线段施工地层损失引起的沉降规律,并与宁波市潘桥变电站顶管隧道工程地表沉降实测值进行对比验证。研究结果表明:1)考虑顶管施工引起的附加荷载、水平与竖向超挖间隙等因素所构建的S形空间曲线顶管施工引起的地表沉降计算方法,理论计算值与实测值吻合较好,能够对地表沉降进行较为准确的预测;2)空间曲线顶管施工引起的地表沉降槽不以顶管中心轴线对称,在顶管机头无超挖刀参与掘进的情况下,最大沉降值向弯道外侧偏移,且顶管机平面转弯产生的超挖间隙对沉降槽偏移影响最大;3)在S形空间曲线顶进时,地层损失是造成地表沉降的主要原因,应严格控制注浆参数,理论注浆量应包括环形空隙与超挖间隙注浆量体积之和。

New versions of Barbalat's lemma with applications

This note presents a set of new versions of Barbalat＇s lemma combining with positive （negative） definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple example shows the usefulness of our results.

A version of Hill's lemma for Cosserat continuum

On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.

The Extension of Robinson's Sequential Lemma and Its Application

A. Robinson＇s sequential lemma is extended to nets in general topological space, and obviously the case of nets in ^＊R is its corollary. As its application, the paper proves a property about topology of uniform convergence.

Union Jack晶格上混合自旋-1/2—自旋-S伊辛模型临界温度的自由费米近似解

考虑Union Jack晶格上各向异性二体耦合相互作用的混合自旋-1/2—自旋-S伊辛模型,运用自由费米近似方法对模型进行了求解,得到了模型临界温度的自由费米近似解.

Wiener's lemma:localization and various approaches

Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.

Approach to a Fifth-Order Boundary Value Problem, via Sperner's Lemma

We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.

Fuzzy Zorn’s lemma with applications

We introduced the fuzzy axioms of choice,fuzzy Zorn’s lemma and fuzzy well-ordering principle,which are the fuzzy versions of the axioms of choice,Zorn’s lemma and well-ordering principle,and discussed the relations among them.As an application of fuzzy Zorn’s lemma,we got the following results:(1)Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal.(2)Every nonzero ring contained a fuzzy maximal ideal.(3)Introduced the notion of fuzzy nilpotent elements in a ring R,and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R.(4)Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma,we proved the fuzzy Tychonoff Theorem.

The Destruction of a Genius——Comment on Jack London's autobiographical novel"Martin Eden

Living a dynamic life physically,intellectually and politically, Jack London had a career that was representative of the vigor of his time with working-class origins, he was, in some of his literary and political writing,one of the most articulate and militant spokesmen of his class. His work is absent from many American authologies,but included in some. "Martin Eden", London’s most autobiographical novel,is considered by many critics to be his most mature work which is read all over the world.

基于T-S故障树的数控加工中心液压托架升降系统可靠性分析

液压系统可靠性分析主要有故障树、故障模式影响与致命性分析、贝叶斯网络等方法,由于历史数据的缺乏及系统工作环境的变化,使得部件故障发生概率为精确值的假设难以成立,而模糊理论的引入,有效解决了这一问题。通过分析THMC6350双工位卧式加工中心液压托架升降系统工作原理,构建了其系统失效时的Takagi-Sugeno(T-S)模糊故障树模型,解决了故障数据缺乏及故障机理不确定问题,进而对系统进行可靠性分析,求得液压托架升降系统在不同故障状态时的故障概率、各底事件概率重要度和关键重要度,为设备故障诊断提供可靠依据。

局部引理及其在(r，s)-SAT问题中的应用

一般说来,寻找满足一定结构性质的对象结构是困难的。概率方法提供了解决此类问题的途径:证明满足一定结构性质的对象的概率大于零。在概率方法中,局部引理是一个关键技术。本文介绍了局部引理的基本原理和使用方法,并将其应用到估计(k,s)-SAT问题中临界函数的下界。

软岩隧道初支拱架S形变形原因分析及对策

围绕云桂铁路软弱隧道的初支"S"形变形,通过力学、支护、监测与数值模拟对比分析,研究了隧道初期支护中钢拱架"S"形变形的原因,得出围岩应力分布不同时,必然会导致不均匀变形来释放应力,钢拱架连接钢板以上50 cm^150 cm范围内,钢架刚度不足会呈现"S"形变形的结论。

多项式环Z_(p^e)[x]中的Hensel引理及提升

在多项式环 Zpe[x]中 ,建立了 Hensel引理及提升 ,并利用 Hensel引理证明了 xn- 1在 Zpe[x]中可惟一分解成基本不可约多项式的乘积 ,其中 (n,p) =1。

S型空间曲线圆形顶管管道应力分析——以宁波市电力隧道顶管工程为例

为探索顶管在S型曲线顶进过程中管道结构荷载变化,以宁波电力顶管隧道工程为背景,结合施工参数对管道环向、纵向钢筋应力和管土接触应力的影响进行实测分析,建立三维数值模拟进行对比,同时采用多工况模拟进行拓展分析。分析结果表明:1)环向钢筋应力分布不均匀且受转弯影响较小,受注浆压力影响较大,管顶和管底外侧受拉、内侧受压,管腰则相反;纵向钢筋应力主要受顶力影响,在S型曲线上管腰应力交替波动,管底内外侧均受拉,其余部位外侧受拉、内侧受压。2)顶管转向产生不均匀的管土间隙,使注浆压力不均匀,纠偏使管道处于泥浆套的右下方,实测管土接触应力大小为管道左侧>管顶≈管底>管道右侧,模拟管土接触应力与实测数据有着相似的变化趋势,应力分布不均匀,且管顶与管底受转弯影响较小,管腰受转弯影响较大。3)拓展模拟得出,管道埋深和坡度较大时,管顶和管底管壁应力变化尤为显著;适当增加注浆压力,可缓解管壁应力不对称、不均匀现象;水平纠偏时纠偏程度不宜过大,避免产生管道应力集中的现象;增大顶力不能使应力重分布,且纵向应力增加不大。