应用a~3 b~3 c~3=3abc解题

常见到三数和为零,求三数立方和的问题。这时应用若a b c=0,则a~3 b~3 c~3=3abc解题非常精炼。下举几例。例1 若a b c=0,a~3 b~3 C~3=0,求a~(1991) b~(1991) c~(1991)的值。解∵a b c=0, ∴a~3 b~3 c~3=3abc=0, ∴a、b、c中有一个为零,另两个互为相反     

a~3 b~3 c~3-3abc的综合应用

在因式分解当中,有一个很重要的公式：a~3 b~3 c~3-3abc=(a b c)(a~2 b~2 c~2- ab-bc-ca)．在做一些复杂题时,往往能因为它而化难为易,化复杂为简单．它的特殊之处在于两点：①当a b c=0时,a~3 b~3 c~3=3abc．②当a b c≠0,a~3 b~3 c~3=3abc时,     

a~3 b~3 c~3≥3abc的几种证法

一个人数学基础是否扎实,主要看他对一些基本定理的理解.然而,同学们往往忽视命题的证明过程,只记其结论,这未免有些不好.在自学中,我对a3 b3 c3≥3abc这一定理发掘出几种证法.     

不等式x~3 y~3 z~3≥3xyz的几何证法

我们先证ｘ２＋ｙ２≥２ｘｙ（ｘ、ｙ∈Ｒ＋,当ｘ＝ｙ时,等号成立）证明　如图１,设正方形ＡＢＣＤ的边长为ｘ,正方形ＢＥＦＪ的边长为ｙ,在ＡＢ上取ＡＨ＝ｙ,则ＨＢ＝ｘ－ｙ,故ＨＥ＝ＨＢ＋ＢＥ＝ｘ－ｙ＋ｙ＝ｘ,∴　Ｓ矩ＡＨＰＤ＝Ｓ矩ＨＥＦＫ＝ｘｙ．由图１显然有　Ｓ正ＡＢＣＤ＋Ｓ正ＢＥＦＪ≥Ｓ矩ＡＨＰＤ＋Ｓ矩ＨＥＦＫ,即　　　ｘ２＋ｙ２≥２ｘｙ（当且仅当ｘ＝ｙ时,等号成立）再证　ｘ３＋ｙ３＋ｚ３≥３ｘｙｚ（ｘ、ｙ、ｚ∈Ｒ＋,当且仅当ｘ＝ｙ＝ｚ时,等号成立）证明　如图２,设三个正方体ＶＡＢ、ＶＣＤ、ＶＥＦ…     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

环F_3+uF_3+vF_3+uvF_3上的循环码

研究了环R=F_3+uF_3+vF_3+uvF_3上循环码的结构(u~2=u,v~2=v,uu=uu),证明了该环上的循环码是主理想生成的,并给出了其上循环码的生成多项式.     

不等式a^3＋b^3＋c^3≥3abc的新证法

对于三元基本不等式“若ａ ,ｂ ,ｃ∈Ｒ+,则ａ3+ｂ3+ｃ3≥ 3ａｂｃ” ,老教材是利用因式分解的办法 ,将ａ3+ｂ3+ｃ3- 3ａｂｃ化为12 (ａ +ｂ +ｃ) [(ａ -ｂ) 2 +(ｂ -ｃ) 2 +(ｃ -ａ) 2 ]后 ,再判断其值的正负而获证的 ,新教材是利用构造的办法 ,联想构造不等式“若ａ ,ｂ∈Ｒ+,则ａ3+ｂ3≥ａ2 ｂ +ａｂ2 ” ,后利用二元基本不等式“若ａ ,ｂ∈Ｒ+,则ａ +ｂ≥ 2ａｂ”而证得的 .显然老教材中的证明对因式分解要求较高 ,学生较难掌握 ,故老教材中的证明被新教材中的证明取而代之了 ,但新教材中的构造证法技巧性亦较强 ,且构造的是一个一般性…     

L'équation ξ3 + η3 + ζ3 = 0 et la courbex 3+y 3=1

Sans résumé     

3×3实矩阵的Volterra乘子

本文应用代数知识,详细讨论3×3实矩阵A存在Volterra乘子D的充分必要条件,以及A的Volterra乘子D唯一的充分必要条件,并且确定A的Volterra乘子中各元素的最小存在范围及具体表达式(当D存在唯一时)。     

The denormalized 3×3 lemma

The classical 3×3 lemma holds in any regular protomodular category with a zero object. It is investigated here whether there is a “denormalized” version when the category no longer has a zero object, as, for instance, any slice category of the category , or any slice category of an abelian category. The answer is actually positive in the weaker context of regular Mal'cev categories.     

Irreducible Representations of Quantum 3 × 3 Matrices

The irreducible representations of quantum 2×2 and 3×3 matrices at the roots of unity are classified.     

D-modules associated to 3×3 matrices

We classify regular holonomic $\text{D}$-modules whose characteristic variety is contained in the union of conormal bundles to the orbits of the group of invertible matrices. The main result is an equivalence between the category of such $\text{D}$-modules and the one of graded modules of finite type over a Weyl algebra. To cite this article: P. Nang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

(3, 3)-Ary Differential Rings

The main purpose of this paper is to introduce the concept of (3, 3)-ary differential rings as a generalization of differential rings. Then we study ideals, prime ideals, radical and nil ideals of them. Finally, we give a construction of hyperstructures determined by (3, 3)-ary differential rings.     

3D Euler spirals for 3D curve completion

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curve, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of this curve - properties that have been shown to be important for the appeal of curves. We illustrate its utility in two applications. The first is “fixing” curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.     

SL(3,3~n)和SU(3,3~n)的第一Cartan不变量

确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.     

3-LIE BIALGEBRAS

3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.     

An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

The Ramsey number N(3,3,3,3; 2) is the smallest integer n such that each 4-coloring by edges of the complete graph on n vertices contains monochromatic triangles. It is well known that 51 ≤ N(3, 3, 3, 3; 2) ≤ 65. Here we prove that N(3, 3, 3, 3; 2) ≤ 64.     

3-Lie bialgebras

3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.