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独立随机变量序列重对数律的一个注记 总被引:1,自引:0,他引:1
{X_i}为独立随机变量序列,E(X_i)<+∞,E(X (2)_(i))<+∞(i=1,2,…),当中心极限定理中的余项△n=O(ln Bnln ln Bn…(lnk Bn)~(1+δ)~(-1))时,本文得出结论: 相似文献
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我们看两类函数(1){af(n)} (n=1,2,…)(2){f(an)} (n=1,2,…)如果数列(1)、(2)是等差(比)数列,那么我们把它们称为复合等差(比)数列.于是,af(n)=af(1)+(n-1)d或af(n)=af(1)qn-1.例1 数列{an}满足2S2n=2anSn-an(n≥2),a1=2,求an及Sn.解 将an=Sn-Sn-1(n≥1)代入等式,得 2SnSn-1=Sn-1-Sn.因为a1=2≠0,故Sn≠0,上式可变为1Sn-1Sn-1=2,∴ 数列{1Sn… 相似文献
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定理设f(x)=a1sin(x+α1)+a2sin(x+α2)+…+ansin(x+αn)(或f(x)=a1cos(x+α1)+a2cos(x+α2)+…+ancos(x+αn))(ai,αi是常量,i=1,2,…,n).如果对x1,x2(x1-x2... 相似文献
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自然数方幂求和问题:即Sp=1p+2p+…+np求和,两千多年来,为人们关注和熟知.三百多年前,贝努利用二项式定理及递归方法,对每个自然数p,可逐个求出Sp.今天,Sp的求法仍在不断被改进、创新.这在许多著作及刊物中均可找到.我们知道:p<-1时,Sp收敛.例如熟知 limn→∞(112+122+…+1n2)=π26.当p≥-1时,Sp发散.(p=-1时Sp=11+12+…+1n,即调和级数,可用递归型公式求和).当p为非负整数时,熟知S0=n,S1=n(n+1)2,S2=16n(n+1)(2n… 相似文献
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利用数列{an}的如下两类变换:an=a1+(a2-a1)+(a3-a2)+…+(an-an-1)及an=a1a2a1a3a2…anan-1(ai≠0,i=1,2,…,n-1)不仅能简便地推导出等差数列和等比数列的通项公式,而且灵活运用它们还能简捷、... 相似文献
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文[1]将Popoviciu不等式修正为:“设xi,yi≥0(i=1,2,…,n),且xp1-∑ni=2xpi>0和yp1-∑ni=2ypi>0,其中0<p≤2,则(xp1-∑ni=2xpi)(yp1-∑ni=2ypi)≤(x1y1-∑ni=2xiyi)p①当且仅当p=2且x1y1=x2y2=…=xnyn时,①式取等号”.这里,应加上“当0<p≤2,x2=x3=…=xn=y2=y3=…=yn=0时,①也取等号”才完整.本文我们将不等式①进一步推广为:定理 设xij>0(i=1,2,…,m,j=1… 相似文献
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两个不等式的简捷证法 总被引:1,自引:0,他引:1
下面给出的两类不等式问题,一般是通过代换的方法证明.本文给出直接简捷的证明.命题1 设xi∈R+(i=1,2,…,n)且x211+x21+x221+x22+…+x2n1+x2n=a(0<a<n),求证:x11+x2+x221+x22+…+x2n1+x2n≤a(n-a)①证 由题设易知:11+x21+11+x22+…+11+x2n=n-a.由于 11+x2k+n-aa·x2k1+x2k ≥211+x2k·n-aa·x2k1+k2k =2n-aa·xk1+x2k)(k=1,2,…,n),此n式相… 相似文献
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Suppose we are given three disjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the given three, and R.M. Krause has shown that a certain alternating sum of the curvatures of these eight circles must vanish. We express this result in an inversively invariant way and determine the extent to which it generalizes to other configurations of three given circles. 相似文献
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We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line
and three spheres for which there are infinitely many lines tangent to the
three spheres that also meet the given line.
All such configurations are degenerate.
The path to this result involves the interplay of some beautiful and
intricate geometry of real surfaces in 3-space, complex projective algebraic
geometry, explicit computation and graphics. 相似文献
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《Mathematical and Computer Modelling》2007,45(1-2):137-148
A natural 3D extension of the Steiner chains problem, original to the authors of this article, where circles are substituted by spheres, is presented. Given three spheres such that either two of them are contained in (or intersect) the third one, chains of spheres, each one externally tangent to its two neighbors in the chain and to the first and second given spheres, and internally tangent to the third given sphere, are considered. A condition for these chains to be closed has been stated and the Steiner alternative or Steiner porism has been extended to 3D. Remarkably, the process is of symbolic-numeric nature. Using a computer algebra system is almost a must, because a theorem in the constructive theory in the background requires using the explicit general solution of a non-linear algebraic system. However, obtaining a particular solution requires computing concatenated processes involving trigonometric expressions. In this case, it is recommended to use approximated calculations to avoid obtaining huge expressions. 相似文献
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Malfatti?s problem, first published in 1803, is commonly understood to ask fitting three circles into a given triangle such that they are tangent to each other, externally, and such that each circle is tangent to a pair of the triangle?s sides. There are many solutions based on geometric constructions, as well as generalizations in which the triangle sides are assumed to be circle arcs. A generalization that asks to fit six circles into the triangle, tangent to each other and to the triangle sides, has been considered a good example of a problem that requires sophisticated numerical iteration to solve by computer. We analyze this problem and show how to solve it quickly. 相似文献
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Ronald L. Graham Jeffrey C. Lagarias Colin L. Mallows Allan R. Wilks Catherine H. Yan 《Discrete and Computational Geometry》2006,35(1):1-36
Apollonian circle packings arise by repeatedly filling the interstices
between four mutually tangent circles with further tangent circles.
Such packings can be described in terms of the Descartes configurations
they contain, where a Descartes configuration is a set of four mutually tangent
circles in the Riemann sphere, having disjoint interiors.
Part I showed there exists a discrete group, the Apollonian group,
acting on a parameter space of (ordered, oriented) Descartes configurations,
such that the Descartes configurations in a packing formed an
orbit under the action of this group. It is observed there
exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral
structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group,
the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric
object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth.
Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral
and the curvature x centers of all circles are integral. We show that (up to scale) there are exactly eight different (geometric)
strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale).
We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes
configurations, which is isomorphic
to the Lorentz group O(3, 1). 相似文献
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J. M. Fitz-Gerald 《Journal of Geometry》1974,5(1):15-26
Degenerate cases of the problem of Apollonius, to construct a circle tangent to each of three given circles, are discussed and exhaustively classified for proper circles (finite and non-zero radius). Singular cases are considered, and an outline of the extension of the problem to higher dimensions given. Amusing alternative interpretations of the results are obtained. 相似文献
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Steve Butler Ron Graham Gerhard Guettler Colin Mallows 《Discrete and Computational Geometry》2010,44(3):487-507
An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement
of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration
is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly
created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how
to fill a curvilinear triangle with circles. 相似文献
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Nguyen Mau Nam Nguyen Hoang Nguyen Thai An 《Journal of Optimization Theory and Applications》2014,160(2):483-509
The classical problem of Apollonius is to construct circles that are tangent to three given circles in the plane. This problem was posed by Apollonius of Perga in his work “Tangencies.” The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls, find the smallest Euclidean ball that encloses all the balls in the first collection and intersects all the balls in the second collection. We also study a generalized version of the Fermat–Torricelli problem stated as follows: given two finite collections of Euclidean balls, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection. 相似文献
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Dr. A. Pendl 《Monatshefte für Mathematik》1975,80(4):307-318
Making use of an earlier result of the theory of stripes in three-dimensional conformal spaceM 3 we obtain a moving frame and derivational formulas for one-parameter families of tangent circles inM 3. Avoiding invariant parameters we can set up a bijection of the tangent circles into the osculating circles that preserves the duple ratio. The following section deals with loxodromes on Dupin cyclides. Finally with the aid of a modified stereographic projection we show how to get the Frenet formulas of the euclidian theory of spacecurves from the derivational formulas for tangent circles inM 3. 相似文献
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G. Megyesi 《Discrete and Computational Geometry》2001,26(4):493-497
We prove that there are at most eight lines tangent to four unit spheres in \R
3
if the centres of the spheres are coplanar, but not collinear. This bound is sharp.
Received October 9, 2000, and in revised form April 26, 2001. Online publication October 12, 2001. 相似文献