不等式a2+b2≥2ab的证明

在常规的数学教学中,对不等式a2+b2≥2ab的证明,一般来说都是采用公式推导的方法.这种方法具有一定的抽象性,使得学生的学习缺乏直观感,对不等式的本质意义理解不到位.     

IVlicrolabFC2a澎湃听音感受

仅仅以命名来看，特别是在看过FC10后，很容易让人把FC20联想成这一类别的第二代产品。不过这种猜想不太正确，FC10是一款2．0结构的桌面近场聆听音箱，而FC20却是一款2．1结构的产品。     

网上音乐新势力a2b

互联网上的B2B或B2C你可能听得多,那么a2b你又有否听过呢?其实就算没有听过也不要紧,因为a2b并不是什么电子商务的新模式,而是由美国最大电话及电讯公司AT＆T所开发出来的网上音乐新格式,无论功能及音响质素皆比目前的MP3为佳,因此极可能成为下一代网上音乐的标准。     

网络新音乐a2b

弹指一挥间,MP3不经意地红透了网络一片天.然而,盗版的阴影却也始终伴随在MP3的背后.于是,所有的敌意、对抗和争夺,也都当然地宣称或真正致力于铲除盗版之恶行.在RIAA(美国唱片行业协会)“安全数字音乐倡议”的摇旗呐喊之下,一些技术公司纷纷作出了积极响应,AT＆T(美国电话电报公司)便是其中之一.自70年代中期以来,AT＆T就一直处于音频压缩研究的前沿,并获得了多项有关音频压缩的基础性专利.对于通过Internet安全地发行音乐作品,AT＆T实验室以“知觉编码”为突破点,开发了“MPEG-2 AAC(Advanced Audio Cod-ing,高级音频编码)”标准,一举推出了比MP3压缩率更高、音质更佳且可有效防止盗版的a2b网络音乐新格式.     

An O(k~2n~2) Algorithm to Find a k-Partition in a k-Connected Graph

Although there are polynomial algorithms of finding a 2-partition or a 3-partition for a simple undirected 2-connected or 3-connected graph respectively,there is no general algorithm of finding a k-partition for a k-connected graph G=(V,E),where k is the vertex connectivity of G.In this paper,an O(k^2n^2) general algorithm of finding a k-partition for a k-connected graph is proposed,where n=|V|.     

L2-bounded stability for a stochastic system†

Both types of linear and non-linear systems with random coefficients are treated in this paper. Four theorems are proved by using the stochastic Liapunov functional, which give the sufficient condition in matrix form for exponential asymptotic stability in the largo with probability one and L₂-bounded stability with probability one.     

BS-2a的中继器故障原因和措施

日本实用广播卫星(BS-2a)于1984年1月23日发射,2月15日进入东经110度静止卫星轨道。卫星在轨道上已运行了一年零九个月,这段时间,卫星出现两个故障,即姿态控制系统单脉冲射频敏感器故障和中继器故障。前者,通过切换到备分系统已经得以解决,后者至使卫星只有一条广播通道能进行电视转播,问题“严重”,虽尽了最大努力,尚未解决。     

BS-2a的三轴姿态建立过程

日本广播卫星BS-2a于1984年1月23日用N-Ⅱ火箭5号在种子岛宇宙中心发射,1月26日远地点发动机点火后进入漂移轨道。1月27日由消旋开始进行一系列三轴捕获工作。经过地球捕获,太阳电池板展开,偏航捕获,飞轮捕获建立了三轴姿态。     

Design of a modified repetitive-control system based on a continuous–discrete 2D model

This paper concerns the problem of designing a modified repetitive-control system for a class of strictly proper plants. Repetitive control involves two types of actions: control and learning; but the insertion of a low-pass filter in a modified repetitive controller, which is employed to guarantee the stability of the system, mixes the two actions together. In this paper, a continuous–discrete two-dimensional model is first constructed. Next, the continuity of repetitive control and Lyapunov stability theory are applied to the model to establish two linear-matrix-inequality (LMI) based sufficient stability conditions, one for the design of the cutoff angular frequency and one for the design of the feedback gains. The features of these conditions are exploited to develop an iterative algorithm that searches for the best combination of the maximum cutoff angular frequency of the low-pass filter and the feedback gains. A numerical example illustrates the design procedure and demonstrates the validity of the method.     

BS-2a的姿态控制和辅助推进系统简介

日本宇宙开发事业团用 N-Ⅱ型火箭5号发射了实用广播卫星 BS-2a。关于这颗卫星的发射情况及其出现的故障已先后在本刊1984年第1期和第4期做过介绍,现对 BS-2a的姿态控制系统和辅助推进系统的功能和性能参数介绍如下:     

Analysis of a 0.1-μm stepping bi-axis piezoelectric stage using a 2-DOF lumped model

Piezoelectric stages are widely used for parts manipulation, including micro-component assembly and bio-cell manipulation. This paper proposes a bi-axis piezoelectric stage based on a friction driven mechanism. The carrier slides on a supporting frame and slider constructed in the same plane as a low-profile stage. The dynamic model of the friction driven is based on a 2-DOF lumped model considering the impact and separation behavior. The first mass is the bulk piezoelectric actuator (slab)—as a vibrator—and the second is the driven slider. The stiffness ratio of the piezoelectric slab-to-the driven slider affects the output force of the slider because of the contact time ratio. Moreover, the stepping size is adjusted according to the duty ratio of the pulse width of the driving signal. The stepping size of 0.1 μm is achieved in the X and Y axes corresponding to 3% and 5% of the duty ratio, respectively, of 512-Hz carrier frequency in 20-Vpp driving signal. When the duty ratio is 100%, the full speed of 18 and 16 mm/s are achieved in the X- and Y-axes, respectively. The dimensions of the piezoelectric stage are 61 × 58 × 21.3 (high) mm³.

     

Bent and hyper-bent functions over a field of 2ℓ elements

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $ P = \mathbb{F}_q We study the parameters of bent and hyper-bent (HB) functions in n variables over a field with q = 2^ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where . The latter has a reduced trace representation F = tr _P ^Q (Φ), where Φ(x) is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case ℓ = 1 to the case ℓ > 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ℓ = 1 these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [1]; we indicate a set of parameters q and n for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics. Original Russian Text ? A.S. Kuz’min, V.T. Markov, A.A. Nechaev, V.A. Shishkin, A.B. Shishkov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 1, pp. 15–37. Supported in part by the Russian Foundation for Basic Research, project nos. 05-01-01018 and 05-01-01048, and the President of the Russian Federation Council for State Support of Leading Scientific Schools, project nos. NSh-8564.2006.10 and NSh-5666.2006.1. A part of the results were obtained in the course of research in the Cryptography Academy of the Russian Federation.     

Cross standard form: a solution to improve a given controller with H 2 or H ∞ specifications

This paper introduces in cross standard form (CSF) as a solution to the inverse optimal control problem. That is, the CSF is a canonical standard problem whose unique H _∞ or H ₂ optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency domain H ₂ or H _∞ specification. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators or two-degree of freedom compensations. Numerical and academic examples are given.     

Fast L1–L2 Minimization via a Proximal Operator

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)–\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)–\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)–\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)–\(L_2\) based on a difference-of-convex approach in the numerical experiments.     

令人困惑的a=*a

“指向二维数组的指针和指针变量”是C语言程序设计中的难点,对于初学者来说有些概念难以理解,容易产生混淆;在实际编程时,也容易错误地加以应用。本人在教学实践中经过摸索发现,从容易理解的一维数组出发,循序渐进地学习与理解此部分内容,就会收到良好的学习效果。     

Performance bounds for nonlinear systems with a nonlinear ℒ2-gain property

Nonlinear ?₂-gain is a finite gain concept that generalises the notion of conventional (linear) finite ?₂-gain to admit the application of ?₂-gain analysis tools of a broader class of nonlinear systems. The computation of tight comparison function bounds for this nonlinear ?₂-gain property is important in applications such as small gain design. This article presents an approximation framework for these comparison function bounds through the formulation and solution of an optimal control problem. Key to the solution of this problem is the lifting of an ?₂-norm input constraint, which is facilitated via the introduction of an energy saturation operator. This admits the solution of the optimal control problem of interest via dynamic programming and associated numerical methods, leading to the computation of the proposed bounds. Two examples are presented to demonstrate this approach.     

ε-Guarantee of a covering of 2D domains using random-looking curves

The problem of covering a given 2D convex domain D with a C¹ random-looking curve C is considered. C within D is said to cover D up to ϵ > 0 if all points of D are within ϵ distance of C. This problem has applications, for example, in manufacturing, 3D printing, automated spray-painting, polishing, and also in devising a (pseudo) random patrol-path that will visit (i.e. cover) all of D using a sensor of ϵ distance span. Our distance bound approach enumerates the complete set of local distance extrema, enumeration that is used to provide a tight bound on the covering distance. This involves computing bi/tri-normals, or circles tangent to C at two/three different points, etc. A constructive algorithm is then proposed to iteratively refine and modify C until C covers a given convex domain D and examples are given to illustrate the effectiveness of our algorithm.     

BS-2a进入静止轨道时辅助推进剂的消耗情况

日本宇宙开发事业团于1984年1月23日发射了实用广播卫星 BS-2a,同年2月15日进入同步轨道,卫星采用三轴零动量姿态控制,卫星的辅助推进系统采用肼反作用方式。推进剂贮箱内携带的是高纯度无苯胺肼(含苯胺量低于0.005%),贮箱压力为350绝对压强(磅/英寸~2),吹下比为1/4,贮箱的最大容量为56公斤,实际上这颗卫星上只携带了52.49公斤肼。卫星从转移轨道到进入静止轨道这段时间的姿态控制和轨道