Building New Functions from Old

The functions we have considered so far, such as x ,x10, 10x, or logx, can be thought of as building blocks out of which we construct other,more complicated,functions. In a simple case,we can take the power functions x and x2 and the constants 3,4, and -5 to create a new function f(x) = 3x2 4x-5. Thus we have produced a quadratic function as a linear combination of power functions. In fact,we have already seen that any polynomial can be thought of as a linear combination of power functions. In this section,we investigate in more de-     

Building,Amalgam和关联几何(II)——Building和关联几何

这里将要介绍的Building和关联几何的主要思想和不少理论都是由Tits提出和创立的.Tits最初提出Building这个几何结构是为了对例外复李群的存在加以几何解释.1955年Chevalley发表了李型群的开创性文章[9].文章中给出了一个一般性方法,使得对每一个复的单李代数和任何一个域,都构造出一个相应的李型群.这样一来,任意域上的例外李型群都获得了一个几何解释.但是,Tits所提出的Building这个几何结构对于研究相应群的各种性质仍然具有基本的意义,从而在有限李型单群的结构理论和表示论、代数群及其表示等方     

Building,Amalgam和关联几何（I）：BN偶与Amalgam

序言 1980年2月,有限单群分类定理宣告“完成”了。几十年以来,围绕着这一定理(或称领域),形成了一支数百人的强大队伍,共中数以十计的人可以毫无愧色地跻身于“世界级”数学家之林。随着这一定理的完成,这支浩浩荡荡的队伍也发生了分化,其中一部分人为简化当时长达万余页的证明而呕心沥血;还有一部分人则利用分类定理及在分类过程中所产生的新思想和新方法,去证明一些历史上悬而未决的难题,或者去开拓新的领域。这两个     

Making Sense by Building Sense: Kindergarten Children’s Construction and Understanding of Adaptive Robot Behaviors

This study explores young children’s ability to construct and explain adaptive behaviors of a behaving artifact, an autonomous mobile robot with sensors. A central component of the behavior construction environment is the RoboGan software that supports children’s construction of spatiotemporal events with an a-temporal rule structure. Six kindergarten children participated in the study, three girls and three boys. Activities and interviews were conducted individually along five sessions that included increasingly complex construction tasks. It was found that all of the children succeeded in constructing most such behaviors, debugging their constructions in a relatively small number of cycles. An adult’s assistance in noticing relevant features of the problem was necessary for the more complex tasks that involved four complementary rules. The spatial scaffolding afforded by the RoboGan interface was well used by the children, as they consistently used partial backtracking strategies to improve their constructions, and employed modular construction strategies in the more complex tasks. The children’s explanations following their construction usually capped at one rule, or two condition-action couples, one rule short of their final constructions. With respect to tasks that involved describing a demonstrated robot’s behavior, in describing their constructions, explanations tended to be more rule-based, complex and mechanistic. These results are discussed with respect to the importance of making such physical/computational environments available to young children, and support of young children’s learning about such intelligent systems and reasoning in developmentally-advanced forms.     

Fokker——Planck——Boltzmann方程

本文研究空间均匀的Fokker--Plan     

RESEARCHANNOUNCEMENTS——TheCauchyProblemsandGlobalSolutionstoDeybeSystemandBurgers''''EquationsinPseudomeasureSpaces

In this paper we consider the construction of solutions to the Cauchy problem of Burgers' equationsut-γ△u + u·▽u = 0, t∈R+,x∈R3, (1)u(0,x)= u0(x), x ∈ R3, (2)in pseudomeasure spaces, where γ(?) 0 is a small parameter that plays the role of the viscosity and u = u(t,x) is a velocity-like vector field defined on R+×R3. The initial datum u0(x) is a vector-valued function defined on R3, In one space dimension Burgers' equation is     

О некоторых свойства х частичных сумм рядо в Фурье—Уолша—Пэли

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n _k } _k ^=1/∞ be some increasing convex sequence of natural numbers such that $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty $$ . Then for anyf∈L ^∞(G) $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty $$ . Theorem 2. Let {n _k } _k ^=1/∞ be a lacunary sequence of natural numbers,n _k+1/n _k≧q>1. Then for anyfεL ^∞(G) $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty } $$ . Theorems. Let µ _k =2 ^k +2 ^k-2+2 ^k-4+...+2^α ₀,α ₀=0,1. Then $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered} $$ . Theorem 4. {{S ₂ k(f: 0)} _k ^=1/∞ ,f∈L ^∞(G)}=m. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ .     

数学王子——高斯

高斯是近代数学伟大的奠基人之一 ,被人们誉为“数学王子” .他与阿基米德、牛顿一起被称为历史上最伟大的三位数学家 .高斯出生在一个贫苦的家庭里 ,祖父是农民 ,父亲做短工 .“穷人的孩子早当家” ,高斯从小爱动脑筋 ,如果承认历史上有神童 ,那么 ,高斯就是其中之一 .传说有一次 ,他父亲计算账目 ,小高斯在旁边好奇地看着 ,等父亲好不容易算完后 ,想不到小高斯说 :“爸爸 ,你算错了 ,应该是……” .经过核对 ,果然是小高斯正确 .高斯的父亲又惊又喜 ,本来不想送他上学 ,考虑到他出奇的聪明好学 ,于是在他 7岁的时候送他进了小学 .小高斯经…     

方法—效果—能力

数学教学的根本目的是使学生获得知识,同时发展学生的智力,培养学生的能力。要过河必须有船或桥,要达到教学目的,必须有行之有效的方法。怎样的方法有利于学生获得知识,有利于培养学生的能力呢?的确这是数学教学中不断研究探索的一个课题,也是关系到数学教学提高质量的重大问题,对此笔者根据自己的教学实践谈点浅见。一教学方法以少胜多数学上的解题是学生所学知识的综合运用,是培养学生能力的集中表现。因而,教给学生解题的思考方法是教学的关键所在。教学实践使我体会到:教方法,以少胜多,事半功倍。下面举例谈谈做法。     

Galton——Watson~���ľֲ����޵�һ������

Take any subcritical offspring distribution with bounded support and consider the corresponding Galton--Watson tree. In this short note we condition this Galton--Watson tree on large width and show that the conditioned tree does not converge locally to any random tree with at most one infinite spine.     

反思—发现—再反思—再发现

学习数学的过程是发现问题和解决问题的过程 .要想发现问题 ,首先要思考 .思考的方式很多 ,在解决一个问题后 ,反思就是一种常用的思考方法 ,这种思考是在一定基础上对问题进行比较、深化和提高 ,这样的思考有利于我们优化解决问题的方法 ,培养思维的广阔性 .下面是笔者在教学中遇到的一例 .问题　已知点A( -1,-3 )为圆x2 +y2=4上一定点 ,B、C为圆上另外两动点 ,且∠BAC =3 0°,求△ABC面积的最大值 .分析　这是一个解析几何中的最值问题 ,解决这类问题的常用方法是 :引入参数 ,建立关于面积的目标函数 ,然后再求解 .设立怎样的参数是解…     

Application of an Artificial Neural Network for Modelling the Thermal Dynamics of a Building’s Space and its Heating System

Artificial neural networks (ANNs) have been used for modelling the thermal dynamics of a building’s space, its water heating system and the influence of solar radiation. A multi-layer feed-forward neural network, using a Levenberg-Marquardt backpropagation-training algorithm, has been applied to predict the future internal temperature. Real weather data for a number of winter months, together with a validated building model (based on the building constructions data), were used to train the network in order to generate a mapping between the easily measurable inputs (outdoor temperature, solar irradiance, heating valve position and the building indoor temperature) and the desired output, i.e., the predicted indoor temperature. The objective of this work was to investigate the potential of using an ANN with singular value decomposition method (SVD) to predict the indoor temperature to shut down the heating system controller early for saving the energy consumption for heating inside the building.     

Nonlinear Programming——a Review

Nonlinear programming problems are to minimize general nonlinear functions, possibly subject to some nonlinear constraints. In this paper, we review some recent results on nonlinear optimization.     

F—析取ω —语言

关于ω—语言的讨论,基本上者是从文法与自动机的角度([1],[2],[3]).但近年来,许多人从代数学的角度来研究ω—语言的特性([4],[5]).本文从语言的句法么半群出发.作为[5]中讨论的析取ω—语言的拓广,定义了 f—析取ω—语言,讨论了这类语言的若干特性.     

关于G——对称多项式

本文将对称多项式的基本定理推广到了关于一般群为对称的多元多项式中,并指出了这一推广在构造含有某些指定根的方程方面的应用。     

Demi-linear analysis Ⅱ——demi-distributions

In this paper, we develop the theory of demi-distributions which generalizes the usual distribution theory. In particular, we show that many results on differentiations, Fourier transforms, and convolutions can be generalized to demi-distributions theory.