Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations （NEE（s））, by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptie equation is highly studied and new type solutions and Backlund transformation are obtained. Then （2＋ l ）-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

Exact Sequence and Commutative Diagrams of Semimodule Homomorphism

Throught the introduction of the concept * exact sequence,we set up several satisfied results of commutative diagrams of semimodule homomorphism.

The Mathematical Foundations of General Relativity Revisited

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.

Fibonacci Congruences and Applications

When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences xn. For that purpose, we will use non-deterministic sequences yn. They are transformed using Fibonacci congruences and we will get by this way sequences xn. These sequences xn admit the IID model for correct model.

关于n-纯同调维数的注记

为了研究环与模的n-纯同调维数,通过同调的方法,分别给出了n-纯投射模与n-纯内射模的一些新刻画.作为应用,证明了环R的n-纯投射整体维数不超过1的充分必要条件是R的n-纯内射整体维数不超过1,得到了一个环R的n-纯投射整体维数与R的n-纯内射整体维数的对应关系.

关于Pell方程组x^(2)-39y^(2)=1与y^(2)-Dz^(2)=16的公解

利用同余、递归序列的方法以及Pell方程解的性质,证明了Pell方程组x^(2)-39y^(2)=1与y^(2)-Dz^(2)=16的公解,情况如下:(1)当D=2^(n)(n∈Z^(+))时,方程组只有平凡解(x,y,z)=(±25,±4,0);(2)当D=2p_(1)…p_(s)(1≤s≤4,p_(1),…,p_(s)是互异的奇素数)时,除开D=2×1249,方程组有非平凡解(x,y,z)=(±62425,±9996,±200)外,仅有平凡解(x,y,z)=(±25,±4,0).

关于不定方程x^(3)-1=114y^(2)

不定方程是数论中不可或缺的一个分支,它有着悠久的历史与丰富的内容,其理论和方法在各学科和实际生活中都有广泛的应用。运用同余式、递归序列、平方剩余以及Pell方程的解的性质等初等方法对不定方程x^(2)-1=114y^(2)的整数解进行了讨论。首先利用因式分解将原不定方程分解为8种情形,其次运用转化、取模等技巧对8种情形分别分析,最终得出不定方程x^(2)-1=114y^(2)仅有整数解(x,y)=(1,0)。

丢番图方程x^(3)+1=603y^(2)的整数解

设D含有6k+1型的素因子,其中k是正整数,关于丢番图方程x^(3)+1=Dy^(2)(D>0)的求解一直是数论中未彻底解决的问题之一.利用同余式、递归序列、因式分解法以及Pell方程解的性质及初等数论方法,并结合分类讨论的数学思想,研究了丢番图方程x^(3)+1=603y^(2)在不同情形下的整数解问题,得到了当D=603时,该丢番图方程有且仅有平凡整数解(x,y)=(-1,0),此结论对这一类方程的研究具有一定的借鉴作用.

Gorenstein(L,A)-投射模的稳定性

设R是有单位元的交换环,(L,A)是完备的对偶对.先引入一种相对于完备对偶对(L,A)的Gore nstein同调模类GP(2)L,再研究GP(2)/L的一些性质.最后,借助一些特殊的模类证明GP(2)L与Gorenstein(L,A)-投射模类一致.

复形的Gorenstein(L,A)-内射维数

设(L,A)是一给定的完备对偶对.首先,引入复形的Gorenstein(L,A)-内射维数,给出其刻画,并证明复形的Gorenstein(L,A)-内射维数不超过内射维数;其次,讨论复形的相对上同调和Tate上同调,得到联系绝对、相对、Tate上同调的长正合序列.

形如8k+1、8k-1、8k+3和8k-3(k∈Z)的素数都有无穷多个

尝试利用反证法和分类讨论法等,分别给出了“形如8k+1(k∈Z)的素数有无穷多个”、“形如8k-1(k∈Z)的素数有无穷多个”、“形如8k+3(k∈Z)的素数有无穷多个”和“形如8k-3(k∈Z)的素数有无穷多个”的严格证明。所用知识都是初等数论中的基础知识,仅限于素数、整除、同余和Legendre符号的一些基本性质。为了证明主要结论,还首先推导出了两个很有用的引理。

Categories of exact sequences with projective middle terms

Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra

半模正合列

在陈培慈、周媛兰文章中半模正合列的概念的基础上,定义了半模的短正合列,并把环上模的正合列的一些性质推广到半模范畴中,得到了半模正合列上的"五引理".

i-内射半模

该文给出了i-内射半模的概念及一些较好的特征刻划,并讨论了内射半模与i-内射半模的关系.主要有左R半模E是i-内射半模当且仅当反变函子HomR(-,E)真正合,并通过给出弱内射半模的定义得出左R-半模E是内射半模当且仅当E既是弱内射半模又是i-内射半模.

关于不定方程x^3+1=91y^2

利用同余式、平方剩余、递归序列、Maple小程序及Pell方程解的性质,证明了不定方程x3+1=91y2仅有整数解(x,y)=(-1,0),(4367,±30252).

p-内射半模

给出p-内射半模的概念,并在此基础上刻划了p-内射半模与Hom函子的关系.接下来证得p-内射半模的直积与直和仍保持是p-内射半模。以及讨论了在无零因子半环中,p-内射半模与可除半模的关系。最后由于p-内射半模定义的条件比内射半模弱,证明了在任意真半环上存在非零的p-内射半模。

i-内射半模的Hom函子刻划

给出i-内射半模的定义后,通过讨论Hom函子是否保持真正合列与短真正合列,得到了i-内射半模的一个重要等价刻画,即把环模的相应结果成功地推广至i-内射半模上。

关于不定方程x^3±1=1455y^2的一个初等解法

利用同余理论、递归序列,以及Pell方程解的性质证明了不定方程x3-1=1 455y2有整数解(x,y)=(1,0),(4 366,±7 563);而不定方程x3+1=1 455y2仅有整数解(x,y)=(-1,0).

不定方程x^3+1=301y^2的整数解

利用递归数列、同余式、平方剩余以及Pell方程解的性质证明不定方程x3+1=301y2仅有整数解(x,y)=(-1,0).