关键词： 数学算法   教学内容   高中   数学教师   北京市   必修课  

摘要： 2007年北京市高一全面实施了新课程，算法作为数学必修课的新增内容纳入了教学视野。由于不少数学教师没有系统学习过算法内容，因此上算法初步课明显的底气不足。现结合算法初步教学内容，针对高中数学教师现状，提出以下建议。

关键词： Jacobi algorithm   Cartan decomposition   Restricted-root space   Structured eigenvalue decomposition   Structured singular value decomposition   Local quadratic convergence   Exceptional Lie algebra  

摘要： A structure preserving sort-Jacobi algorithm for computing eigenvalues or singular values is presented. It applies to an arbitrary semisimple Lie algebra on its (-1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as e.g. the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. (C) 2008 Elsevier Inc. All rights reserved.

关键词： 算法   数学思想   程序方法   程序化  

摘要： 一、程序化思想 算法是以问题为载体，是解决问题的一种程序方法，它通常指向某一个或某一类问题．算法案例是算法程序化思想的体现．

关键词： 细胞图像   分割算法   数学形态学   分水岭算法  

摘要： 图像分割是由图像处理进入到图像分析的关键步骤,是图像工程中的一个重点和难点。目前关于图像分割已有很多成熟的算法,不同算法都有各自的优点和不足之处。由于细胞图像的特殊性,目前的分割算法研究结果在实际处理中,经常会出现多个细胞粘连或者重叠在一起的现象,严重影响了后续的统计分析。\n 基于数学形态学的分水岭算法是一种有效的图像分割方法,但是传统的分水岭算法存在严重的过分割现象。通过合并预处理步骤来限制允许存在的区域的数目是解决过分割问题的一个有效办法;另外一个解决的办法是使用标记,使用形态学中腐蚀运算对细胞图像进行连续腐蚀得到分离细胞种子。考虑到人体血液红细胞具有类圆形的特征,使用固定的结构元素腐蚀容易导致细胞图像失真,不利于后续的图像分析工作。使用不同的结构元素对细胞图像进行交替腐蚀则可以有效避免以上问题。\n 本文首先介绍了几种常用的重叠细胞分割方法,然后研究了一种基于形态腐蚀和膨胀运算的,应用分水岭算法原理对重叠红细胞图像分割的改进算法。该算法首先使用4邻域和8邻域结构元素对重叠细胞图像连续交替腐蚀,通过计算腐蚀后的细胞面积与原细胞面积的比值来控制连续腐蚀次数。在获取的分离细胞图像基础上,应用分水岭算法原理构造重叠细胞之间的分割线。尽管腐蚀和膨胀不是一对可逆运算,但是使用交替结构元素可以减少类圆形红细胞因连续腐蚀和膨胀运算而造成的失真。该算法主要针对红细胞具有类圆形的特征设计的,在本文的第五章中给出了使用该算法对重叠细胞图像分割的实验结果。从实验的结果来看,绝大部分重叠细胞被分离,从而验证了本文算法对重叠红细胞图像分割具有良好的效果。并且该算法可以推广应用于一些类圆形目标的图像分割上。

关键词： 高中教育   数学课程   算法化教学   学习策略  

摘要： 笔者针对2004年开始的课程改革试点，并结合在实践中积累的大量教学经验，在广州市执信中学展开有关提高课堂有效性的教学实验。本实验研究通过“算法化教学”微型实验，实证“算法化教学”法的可行性与必要性，努力做到从经验描述水平提高到理论假设水平，从自然观察水平上升到实践思辨水平，从追求成果表述扩大到交流传播并物化为教学质量的提高。\n 这个教学法的特点是重视数学思想在教学过程的渗透，重视专家演练与自主学习的有机结合，教学方式博采众家之长，不拘一格，体现“教无定法”;在教学的过程中形成融洽的师生关系，能者为师，在引导学生探索数学世界中，教师身担多重角色（如执教者、引导者、咨询者或资料提供的伙伴）;注重学生“数学化”的形成，注重学生人格的发展;教学理念不偏颇一方，对各种已有的教学模式及理念尽力做到“执两端而用之”，客观、公正地看待每一种对教育有过重要意义的观点，根据自身特点，古为今用（如算法化的强调），洋为中用（自主学习、研究性学习等方式），形成一套有特色的教学法。\n 实证结果表明:算法化教学法有效激发学生的学习兴趣，大面积提高学生的数学学业成绩，改善学习方法，对其他学科的学习产生正迁移，有效提高学习的效率。

关键词： 18O内标法   定量蛋白质组学   数学算法  

摘要： 18O内标蛋白质质谱定量分析是一种极具前景的比较蛋白组学研究技术,虽然该方法与迄今所有内标法相比有一系列优点,但是由于其质谱是由16O未标记品种,单及双18O标记品种分子同位素峰簇相互迭加所构成的复杂谱,定量分析的关键是发展有效的质谱解析算法。有鉴于此,综述和讨论了18O内标质谱差异比较蛋白质分析过程中,分子同位素簇分布的简化计算方法,以及质谱混和谱解析的数学方法。

关键词： Linear algebra   Mathematical software   High performance computing   Multicore  

摘要： As multicore systems continue to gain ground in the high performance computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents algorithms for the Cholesky, LU and QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in out of order execution of tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with LAPACK algorithms where parallelism can only be exploited at the level of the BLAS operations and vendor implementations. Published by Elsevier B.V.

关键词： Piecewise algebraic curve   Multivariate spline   Truncated sign   Groebner bases   Interval arithmetic  

摘要： Piecewise algebraic curve is defined as the zero set of a bivariate spline. In this paper, we mainly study the intersection points algorithm for two given piecewise algebraic curves based on Groebner bases. Given a domain D and a partition Delta, we present a flow and introduce the truncated signs, and then represent the two piecewise algebraic curves in the global form. We get their Groebner bases with respect to a lexicographic order and adopt the interval arithmetic in the back-substitution process, which makes the algorithm numerically precise. An example is also presented to show the algorithm's feasibility and effectiveness.

关键词： pattern recognition   estimation algorithm   matrices of estimates   correct algorithm   normalization   algebra over algorithms  

摘要： Algebra over estimation algorithms with addition, multiplication by a constant, and normalization operations is investigated. Normalization is interpreted as a linear (with respect to each row) transformation of the matrix of estimates that takes the maximum entry of the row to unity and the minimum entry to zero. The algebraic closure is described, a formula for its dimension is obtained, and correctness criteria are formulated.

关键词： Dihedral Angle   Combinatorial analysis  

摘要： Geometric Problems become increasingly intractable and difficult to visualize as the number of dimensions increases beyond three. Inductions from lower dimensional spaces are possible yet often awkward. This thesis shows how elementary linear algebra, vector calculus, and combinatorics offer improved methods for calculating the dihedral angles of n-simplicies and proving a generalization of the Pythagorean theorem directly. A large class of search problems are solved in O(log n) time with a traditional binary search. This thesis extends this to a class of problems where the cost of high guesses differs from the cost of low guesses. Weighted binary searches have broad applicability, and in particular, the bulb drop problem is extended, analyzed and solved. Modern Rubik cube research focuses on solving the cube as fast as possible using heavy machinery and computation. Speed cubists routinely memorize 115 different move sequences and practice relentlessly so that they might solve the cube in 15-20 seconds. Mathematicians apply abstract algebra to perform massively parallel computations in order to seek out the fastest way to solve the cube from any position. The diameter of the Rubik’s cube group remains undetermined as of this writing; recently, God’s Algorithm, the method of solving the cube in the fewest possible moves from a scrambled position, was given a least upper bound of 26 moves. This thesis considers the problem from the opposite end of the spectrum. What simple classes of move sequences exist that solve the cube? Since every position of the cube can be achieved by making quarter turns of the faces, quarter turns is a class of moves that solves the cube. It can also be shown that half turns is insufficient. This thesis shows that commutators of quarter turns suffices using methods that have applicability to other cube related problems. There are many ways that the permutation groups may be mapped onto the integers. Some of these maps have interesting prope