1) Cauchy inequality
Cauchy不等式
1.
On the equivalence of the Hoder inequality and the Cauchy inequality;
Hoder不等式与Cauchy不等式的等价性
2.
Generalizations of Cauchy inequality about matrix versions;
Cauchy不等式矩阵形式的推广
3.
Cauchy inequality is very important and widely u sed in inequality studies.
Cauchy不等式在不等式研究中有着重要的地位和广泛的应用。
3) Cauchy-Schwarz inequality
Cauchy-Schwarz不等式
1.
The generalization of Cauchy-Schwarz inequality and its application in matrix analysis
Cauchy-Schwarz不等式的推广及其在矩阵分析中的应用
2.
By using the Cauchy-Schwarz inequality and the properties of octonions,we give an elementary proof of it.
文章利用Cauchy-Schwarz不等式及八元数的一些性质,给出了它的一个初等证明。
3.
By means of induction and analogy,on the basis of analyzing and studying Cauchy-Schwarz dispersed inequality,a new integral popularization of Cauchy-Schwarz inequality has been obtained.
在分析和研究Cauchy-Schwarz不等式的基础上,运用归纳类比的方法,得到了Cauchy-Schwarz不等式的又一个积分推广形式,并给出了一种简洁有趣的构造性的证明。
4) Cauchy-Schwartz inequality
Cauchy-Schwartz不等式
1.
By using the state function it was studied that the influence of the Kerr effect and the initial light field intensity on the two-mode intensity correlation function, the Cauchy-Schwartz inequality, and the second-order correlation function for each mode.
导出了类Kerr介质中双模SU(1,1)相干态场与四能级原子相互作用系统的态函数,研究了Kerr效应和初始光场强度对双模SU(1,1)相干态场的互关联函数、Cauchy-Schwartz不等式及二阶相干度的影响。
2.
The influences of detuning, Kerr medium and initial atomic coherence on the atomic population and Cauchy-Schwartz inequality in the system of SU(1,1) coherent states interacting non-resonantly with a A-type three-level atom with Kerr medium are investigated.
研究了Kerr介质中双模SU(1,1)相干态场与A型三能级原子非共振相互作用系统中场模失谐量、克尔介质以及原子初态对原子布居概率和场的Cauchy-Schwartz不等式的影响。
3.
We find out that the minimum correlation states disobey with the classical Cauchy-Schwartz inequality and exhibit non-classical correlations.
发现在一定的参数范围内双模最小关联混态的二阶相干性违反经典的Cauchy-Schwartz不等式,呈现非经典性相关;同时对双模最小关联混态的压缩特性、亚泊松分布等非经典性质进行了分析,通过数值计算得出,每模光子的压缩性及其亚泊松分布均与参数d的取值密切相关。
5) Buniakowski-Cauchy inequality
Buniakowski-Cauchy不等式
1.
On the basis of analyzing and studying Buniakowski-Cauchy inequality,a new strengthened popularization of Buniakowski-Cauchy inequality s infinite sum is obtained.
在分析研究Buniakowski-Cauchy不等式的基础上,得到了此不等式的无限可和性的新加强推广形式,并给出了十分简洁有趣的构造性方法的证明。
6) Cauchy-Schwarz inequalities
Cauchy-Schwarz不等式
1.
Nonlinear trio-coherent states are introduced and the properties of the completeness relation,number distribution and Cauchy-Schwarz inequalities are studied.
引入了非线性Trio-相干态,讨论了该量子态的完备性及其光子数统计分布和Cauchy-Schwarz不等式。
2.
Let X be an nonnegative random variable and 0<m≤X≤M,converse of two Cauchy-type inequalities are discussed,some moment inequalities of X are obtained as follow:E(X2)-(E(X))2≤14(M-m)2,E(X2)-E(X)≤(M-m)24(M+m),E(X)-(E(X-1))-1≤(M-m)2,E(X)E(X-1)≤(M+m)24Mm,and several converse Cauchy-Schwarz inequalities are also derived.
设X是非负随机变量且0
补充资料：Cauchy不等式
Cauchy不等式
Caudly inequality
Ca”由y不等式「〔渔u曲yin四业‘ty;E加.”epase”e，o] l)关于实数的有限和的Cauchy不等式是指不等式 卧饭{’·却客酷它是由A.L.Cauchy证明的(l 821);关于积分的类似不等式称为E，洲翔以‘不等式(Bunykovs对ine-quality). 2)Cauchy不等式这个名称也用来称呼关于正则解析函数f仁)在复平面c的固定点a上的导数的模Lfk(a)l的一个不等式，或者关于f(z)的幂级数展开式 f(z)=艺ck(:一。广 k=0的系数的模}c*}的不等式.这两个不等式是一f“，(。)j、、!粤，一。}、缪，(·， r、r其中r是使得f(z)为正则的任何圆盘U={:任C:}:一aI簇r}的半径，M(r)是lf(z)}在圆周}:一al二r上的最大模.不等式(*)出现于A.L.Cauchy的著作中(例如见【l]).由这两个不等式可以直接推出Cauchy-Hardamard不等式(Cauchy一Hadamard inequality)(见[2]): f.，‘，、，、.、一/介 lim sup!山气二今二二.}《万二二万不:， 仄而一f Ik!}一d(a，aD)’其中d(a，刁D)是从a到f(z)的全纯域(domain of holo-morphy)的边界沁的距离.特别是，如果f哟是整函数，则在任何点a任C上，有 f.，，;、，、.、1/k hm sup卜一二书且}==0· 荡一r{划! 对于多复变量:=(z:，…，孔)(n>l)的全纯函数f(z)，Cauchy不等式是 1护、++k·刀。、j__M(r，..…。) l‘‘一一-------~‘二二孟l‘二杏，，…奋甲一~-孟一二奋----山监二二 }气_k，气k。l一’一l‘一n’k，k_ }。21’“‘dz矿!rf”‘r矿或 1 IM(r，..…几、 I。奋l(一. 1 lrl’·‘’r矿 “=(口I，…，口，)任(，，，k，，…，人，二0，!.…，其中叭、，…，*。是f(z)的下列幂级数展开式的系数:f(:)二觉ck…k。(:，一。!)“·…(z。一a。)气， kl，..，k。=O其中r，，…，‘是使得f(习为全纯的多圆柱U性{:“C”:lzj一ajI簇rj，j=1，…，‘}的各个半径，M(r1，…，气)是}f(z)!在U”的特异边界上的最大值. 关于参考文献，见Cau由y一H翻心.盯耐定理(Cau-chy一Hadamard theorem). E.八.Cb月o翻r“ueB撰【补注】在西方的文献中很少使用EyF图阳BC班成不等式这个名称，不论是关于实数的有限和的不等式，还是它在复数情况的推广(见双脚，劝.，.曲不等式(B unyako-vs目inequality))，以及关于积分的类似不等式，通常都称为Schwarz不等式(SChwarz inequality)或Cau-chy一schwarz不等术(Cauchy一schwarz inequality)· 上述多圆柱U”的特异边界是集合T”={:“C卜}z，一a、卜rv，，=1，‘’‘，心·张鸿林译
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