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1.
刘健先生在文[1]中提出如下猜想:在任意△ABC中,有cosBcosCsinA2+cosCcosAsinB2+cosAcosBsinC2<1①笔者通过研究,发现了这个不等式的一个指数形式:定理在△ABC中,有cosBcosCsinkA2+cosCco...  相似文献   

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1996年,周永良先生在全国第三届初等数学研究学术交流会论文集中提出如下三角不等式在锐角三角形ABC中,有cos(B-C)cosA+cos(C-A)cosB+cos(A-B)cosC≥6(1)cosAcos(B-C)+cosBcos(C-A)+cos...  相似文献   

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宋发奎 《数学通讯》2000,(10):28-29
我在高三数学培优训练中出了这样一道题 :在△ABC中证明cosAcosBcosC≤ 18. ( 1)学生的解法使我既惊喜又欣慰 .有些解法是我开始没有预料到的 ,有些解法又暴露出同学们对某些数学思想、方法理解中的问题 ,现呈献给大家供参考 .分析 1:积化和差会出现cos (B C )而在△ABC中 ,cos(B C) =-cosA ,从而产生下列解法 :解法 1 cosAcosBcosC= 12 cosA[cos(B C) cos(B -C) ]= - 12 cos2 A 12 cos(B -C)cosA= - 12 [cosA - 12 cos(B -C) ] 2 18cos2 (B…  相似文献   

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解答三角形中的问题 ,除正确运用正弦定理、余弦定理外 ,还应重视它们的两个推论 .利用推论解题 ,方法简捷 ,过程明了 .1 两个推论推论 1 在△ABC中 ,有a =bcosC +ccosB ,b =ccosC +acosC ,c =acosB +bcosA .推论 2 在△ABC中 ,有bcosB +ccosC=acos(B -C)≤a (1 )ccosC +acosA =bcos(C -A)≤b (2 )acosA +bcosB =ccos(A -B)≤c (3 )当且仅当 :B =C时 ,(1 )中等号成立 ;C=A时 ,(2 )中等号成立 ;A =B时 ,(3 )中等号成立 .证 推论 1 :由…  相似文献   

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孙建斌 《数学通讯》2001,(22):28-29
本刊 2 0 0 0年第 2期刊出了《一个不等式的另几种证法与推广》[1] ,读后受益匪浅 .1 一点商榷意见及修正意见文 [1 ]在证明a3 b3 c3≥ 3abc (a ,b ,c∈R )时 ,似有“循环论证”之嫌 .现恭抄其证明过程如下 :    a3 b3≥a2 b ab2 ≥ 2a3b3,    b3 c3≥b2 c bc2 ≥ 2b3c3,    c3 a3≥c2 a ca2 ≥ 2c3a3,三式相加得a3 b3 c3≥a3b3 b3c3 c3a3       ≥ 33 a6 b6 c6 =3abc.在证明过程的最后 ,作者绕了一个大圈 ,最终还是利用a3 b3 c3≥ 3abc来证明a3 b3 c3≥ …  相似文献   

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本文给出了在及c的条件下不可由ax+by+cz表出的最大正整数:b(a,c)十c(a,b)-a-b-c.  相似文献   

7.
在△ABC中 ,记内角A ,B ,C的对边为a ,b ,c,S为△ABC的面积 ,由余弦定理可得 :ctgA =b2 c2 -a24S ,ctgB =c2 a2 -b24S ,ctgC =a2 b2 -c24S .我们称这组公式为“余切公式” .“余切公式”在形式上有与余弦定理相媲美的对称美 ,且在解决三角形中与内角的正 (余 )切有关的命题时 ,能起到化繁为简 ,化难为易之功效 ,请看如下例题 .例 1  ( 1 999全国高中数学联赛试题 )在△ABC中 ,记BC =a ,CA =b ,AB =c ,若9a2 9b2 - 1 9c2 =0 则ctgCctgA ctgB=.解 由余切公式得 :c…  相似文献   

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本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Rucci平坦的流形必定平坦.  相似文献   

9.
在三角形中 ,隐含着一个非常重要的条件 ,而同学们在解题时常常忽略该条件 ,从而造成解题失误 .这个条件就是 :三角形中 ,任意两内角的余弦之和为正 ,即△ABC中 ,cosA cosB >0 ,cosB cosC >0 ,cosC cosA >0 .证明 ∵△ABC中 ,0 <A B <π ,∴ 0 <A <π -B <π ,又余弦函数在 (0 ,π)上是减函数 ,∴cosA >cos(π -B) ,即cosA >-cosB ,故cosA cosB >0 .同理可证 :cosB cosC >0 ,cosC cosA >0 .下面举例说明这一条件的应用 .例 1 已知△ABC中 ,cosA…  相似文献   

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一个三角不等式的改进410005湖南省长沙市一中禹平君众所周知,在△ABC中,有不等式ctgA+ctgB+ctgC≥tg+tg+tg今改进之,我们得到定理在△ABC中,有不等式ctgA+ctgB+ctgC≥tgsec2B、C——AC、A——B十igM...  相似文献   

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We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.  相似文献   

13.
It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.  相似文献   

14.
Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.  相似文献   

15.
As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years (2001-2005). The main topics are: convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.  相似文献   

16.
张丽娜  吴建华 《数学进展》2008,37(1):115-117
One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the'living world. In his seminal paper "The Chemical Basis of Morphogenesis", Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates(1990) on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing's prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows  相似文献   

17.
In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).  相似文献   

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<正>Aims and Scope Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,is one of the transactions of China Society for Industrial and Applied Mathematics,and is a bimonthly journal.JMRA is dedicated to publishing first-rate original research papers in all areas of mathematics with applications,and making research findings available to a wide scientific world,as JMRE has for many years.In line with the name change,the new scope of Journal of Mathematical Research with Applications will not include the articles on mathematical methodology and mathematical philosophy.Copyright Information  相似文献   

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