费波那契数列在珠算教学中的应用

费渡那契,(Fibonacci)数列是13世纪意大利的名数学家费波那契所发明的一种无穷数列．与黄金分割率有莫大的关系,在此我仅就费渡那契数列的基本演算及其在珠算教学中的应用加以论述。     

费波那契数列在珠算教学中的应用

一、前言 费波那契(Fibonacci)数列是13世纪意大利的名数学家费波那契所发明的一种无穷数列,与黄金分割率有莫大的关系,在此我仅就费波那契数列的基本演算及其在珠算教学中的应用加以论述。     

无穷电阻网与黄金分割率

斐波那契数列,又称“兔子数列”.在现代物理、准晶体结构、化学、生物等领域,斐波那契数列都有直接的应用.斐波那契数列有许多奇特的性质,笔者介绍斐波那契数列的黄金分割性质与无限电阻网络的关系.     

斐波那契数列与黄金分割数

对任何固定步长k(≥1),斐波那契数列中相距k项的元形成的子列的前后项之比形成的数列收敛,其极限仅与步长k有关。     

浅谈黄金分割和斐波那契数列

黄金分割和斐波那契数列有着悠久的历史和广泛的应用,很久以来就是初等数学研究,的对象,本文就对这两个著名问题的产生、发展以及若干性质,特别是对两者之间的关系作简单论述。     

卢卡斯数列与斐波那契数列的递归关系研究

法国数学家Edward Lucas曾将数列0,1,1,2,3,4,8,13…命名为斐波那契数,随之而来的则是另外一个数列2,1,3,4,7,11,18…这就是人们所说的卢卡斯数列.卢卡斯数列(下左)与斐波那契数列(下右)有着相同的递归方程,但其首项不同. { Ln+2=Ln+Ln+1L0=2 L1=1 {Fn+2=Fn+Fn+1{F0 =0{F1 =1 事实上,在卢卡斯数列与斐波那契数列中呈现了许多相似的性质.在斐波那契数列中,如果p是q的因子,那么斐波那契数Fp同样是Fq的因子.例如,3是6的因子,那么F3=2也是F6=8的因子.     

一道竞赛题的朴素证法

由递推关系F_(n+2)=F_(n+1)+F_n(n∈N)和F_0=1, F_1=1所确定的数列1,1,2,3,5,8,13,21,34,…叫做裴波那契数列。裴波那契数列有很多性质,在这里只介绍两个下面要用到的(也是常用的)性质: (1)F_nF_(n+2)-F_(n+1)~2=(-1)~n; (2)F_(m+n)=F_mF_n+F_(m-1)F_(n-1)(m≥1,n≥1)。在现实生活中,很多现象与裴波那契数列有关;在数学竞赛中,考裴波那契数列的题目也不少,但对     

广义斐波那契数列

讨论广义斐波那契波数列的定义及其通项表达式,由此可以简单地求出斐波那契数列的通项;同时,讨论广义斐波那契数列的一些应用.     

优美奇异的黄金分割

一、黄金分割 把一条线段分成两段，使其中较大的一段是原线段与较小一段的比例中项，叫做把这条线段黄金分割．在线段AB上取一点P，使得AP：AB—PB：AP（即AP^2=AB×PB），则点P叫做线段AB的黄金分割点．由对称性知，一条线段的黄金分割点有两个P1、P2．     

内蕴斐波那契数列文化的问题

1 斐波那契数列斐波那契数列是意大利数学家LeonardoFibonacci最初发现的,这个数列是:1,1,2,3,5,8,13……,从第三项开始,每一项等于它的相邻前两项之和,用公式表示为Fn=Fn-1 +Fn-2,n=3,4,5,……     

基于黄金分割法的离散回归方程的变点估计及其应用

本文利用变点统计学和黄金分割法讨论有多个变点的离散回归方程的交点估计和参数估计，文中提出基于黄金分割法搜索最佳变点估计和同时得到参数估计的最小二乘算法，还讨论该算法在控制领域的应用，数值模拟结果显示本文算法能给出良好的变点及参数的估计值。     

A certain feature of logarithmic spirals

A curve formed by inversion of a logarithmic spiral is called a double logarithmic spiral. The curves in this family possess the following property: there always exists such a spiral with continuous and monotone curvature satisfying any possible boundary conditions (endpoints, tangents, and curvatures). The problem of constructing a spiral with continuous curvature and prescribed curvature elements at the endpoints is thus solved. Bibliography: 6 titles.     

Polar actions with a fixed point

We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces.     

虽经沧海 依然故我——雅格布·伯努利与对数螺线

简述了雅格布·伯努利(Jacob Bernoulli)的生平及主要贡献,讨论了对数螺线的性质,并给出了对数螺线的一些应用.     

Scale heights of 84 northern spiral galaxies

Using the method proposed by Peng (1988) on the basis of density waves theory and the solution of three-dimensional Poisson’s equation for a logarithmic disturbance of density, and analyzing the spiral patterns, the scale heights of 84 northern spiral galaxies, whose images are taken from the Digitized Sky Survey¹⁾ at Xinglong Observational Station of Beijing Observatory, are measured. The spiral arms of all these galaxies have been fitted on their photographs with some logarithmic spiral curves for getting their correct inclinations.     

Scale height determination of spiral galaxies

The method adopted here is based on the rigorous solution of Poisson’s equation for logarithmic disturbance density within finite thickness galaxies. After their spiral arms are fitted directly with logarithmic spirals, the morphological parameters, scale heights and their relative errors for 32 spiral galaxies, such as NGC4814, are obtained.     

On rational cuspidal curves

Let be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that . We show that where is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When , we show that and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types. Received: 11 February 2000 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502     

Thicknesses of 70 southern spiral galaxies

Using the method proposed by Peng (1988) and analyzing the spiral patterns, the thicknesses of 70 southern spiral galaxies whose images were taken from the Digitized Sky Survey at Xinglong Observational Station of Beijing Observatory were measured. The spiral arms of all these galaxies have been fitted on their images with some logarithmic spiral curves to get their optimum inclinations. Project supported by the Yational Natural Science Foundation of China, the National Climbing Project of China and the Doctoral Program Foundation of State Education Cummission.     

改进鲸鱼优化算法在电力负荷调度中的应用

为了提高鲸鱼优化算法(WOA)的全局优化性能, 提出了一种基于黄金分割搜索的改进鲸鱼优化算法(GWOA)。首先利用黄金分割搜索对WOA的初始种群进行初始化, 使得初始种群能够尽可能的靠近全局最优解, 然后利用黄金分割搜索所形成的变区间, 进行变区间黄金分割非均匀变异操作, 以增加WOA的粒子多样性和提高粒子跳出局部最优陷阱的能力, 从而改善WOA的寻优性能。选取了15个大规模测试函数进行数值仿真测试, 仿真结果和统计分析表明GWOA的寻优性能要优于对比文献的改进鲸鱼优化算法(IWOA)。此外, 将GWOA用于对工程实际应用领域中的电力负荷优化调度问题进行实例分析, 实例应用结果表明, GWOA能有效对电力负荷优化调度问题进行寻优求解。     

Promoting convergence: The Phi spiral in abduction of mouse corneal behaviors

Why do mouse corneal epithelial cells display spiraling patterns? We want to provide an explanation for this curious phenomenon by applying an idealized problem solving process. Specifically, we applied complementary line‐fitting methods to measure transgenic epithelial reporter expression arrangements displayed on three mature, live enucleated globes to clarify the problem. Two prominent logarithmic curves were discovered, one of which displayed the ? ratio, an indicator of an optimal configuration in phyllotactic systems. We then utilized two different computational approaches to expose our current understanding of the behavior. In one procedure, which involved an isotropic mechanics‐based finite element method, we successfully produced logarithmic spiral curves of maximum shear strain based pathlines but computed dimensions displayed pitch angles of 35° (? spiral is ～17°), which was altered when we fitted the model with published measurements of coarse collagen orientations. We then used model‐based reasoning in context of Peircean abduction to select a working hypothesis. Our work serves as a concise example of applying a scientific habit of mind and illustrates nuances of executing a common method to doing integrative science. © 2014 Wiley Periodicals, Inc. Complexity 20: 22–38, 2015