Eigenvectors and Reconstruction

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Document Clustering Using Locality Preserving Indexing

Xiaofei He Department of Computer Science The University of Chicago 1100 East 58th Street, Chicago, IL 60637, USA Phone: (733) 288-2851 xiaofei@
Jiawei Han Department of Computer Science University of Illinois at Urbana Champaign 2132 Siebel Center, 201 N. Goodwin Ave, Urbana, IL 61801, USA Phone: (217) 333-6903 Fax: (217) 265-6494 hanj@
document clustering [28][27]. They model each cluster as a linear combination of the data points, and each data point as a linear combination of the clusters. And they compute the linear coeﬃcients by minimizing the global reconstruction error of the data points using Non-negative Matrix Factorization. Thus, NMF method still focuses on the global geometrical structure of document space. Moreover, the iterative update method for solving NMF problem is computational expensive. In this paper, we propose a novel document clustering algorithm by using Locality Preserving Indexing (LPI). Diﬀerent from LSI which aims to discover the global Euclidean structure, LPI aims to discover the local geometrical structure. LPI can have more discriminating power. Thus, the documents related to the same semantics are close to each other in the low dimensional representation space. Also, LPI is derived by ﬁnding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the document manifold. Laplace Beltrami operator takes the second order derivatives of the functions on the manifolds. It evaluates the smoothness of the functions. Therefore, it can discover the non-linear manifold structure to some extent. Some theoretical justiﬁcations can be traced back to [15][14]. The original LPI is not optimal in the sense of computation in that the obtained basis functions might contain a trivial solution. The trivial solution contains no information and thus useless for document indexing. A modiﬁed LPI is proposed to obtain better document representations. In this low dimensional space, we then apply traditional clustering algorithms such as k -means to cluster the documents into semantically diﬀerent classes. The rest of this paper is organized as follows: In Section 2, we give a brief review of LSI and LPI. Section 3 introduces our proposed document clustering algorithm. Some theoretical analysis is provided in Section 4. The experimental results are shown in Section 5. Finally, we give concluding remarks and future work in Section 6.

Eigenvalues,Eigenvectors

18.03 Class 32, May 1Eigenvalues and eigenvectors[1] Prologue on Linear Algebra.Recall [a b ; c d] [x ; y] = x[a ; c] + y[b ; d] :A matrix times a column vector is the linear combination ofthe columns of the matrix weighted by the entries in the column vector. When is this product zero?One way is for x = y = 0. If [a ; c] and [b ; d] point indifferent directions, this is the ONLY way. But if they lie along a single line, we can find x and y so that the sum cancels.Write A = [a b ; c d] and u = [x ; y] , so we have been thinking about A u = 0 as an equation in u . It always has the "trivial" solution u = 0 = [0 ; 0] : 0 is a linear combination of the two columns in a "trivial" way, with 0 coefficients, and we are asking when it is a linear combination of them in a different, "nontrivial" way.We get a nonzero solution [x ; y] exactly when the slopes of thevectors[a ; c] and [b ; d] coincide: combination of the entries in A "determinant" of the matrix:c/a = d/b , or ad - bc = 0. This is so important it's called thedet(A) = ad - bcWe have found:Theorem: Au = 0 has a nontrivial solution exactly when det A = 0 .If A is a larger *square* matrix the same theorem still holds, withthe appropriate definition of the number det A .[2] Solve u' = Au : for example with A = [1 2 ; 2 1] .The "Linear Phase Portraits: Matrix Entry" Mathlet shows that some trajectories seem to be along straight lines. Let's find them first. That is to say, we are going to look for a solution of the form u(t) = r(t) vOne thing for sure: the velocity vector u'(t) also points in the same (or reverse) direction as u(t). So for any vector v on this trajectory,A v = lambda vfor some number lambda. This Greek letter is always used in this context.[3] This is a pure linear algebra problem: A is a square matrix, andwe arelooking for nonzero vectors number v such that A v = lambda v for somelambda. In order to get all theside asv's together, write the right hand lambda v = (lambda I) vwhere I is the identity matrix [1 0 ; 0 1] , and lambda I is the matrix with lambda down the diagonal. Then we can put this on the left:0 = A v - (lambda I) v = (A - lambda I) vDon't forget, we are looking for a nonzero v . We have just found an exact condition for such a solution:det(A - lambda I) = 0This is an equation in lambda ; we will find lambda first, and then set about solving for v (knowing in advance only that there IS a nonzerosolution).In our example, then, we subtract lambda from both diagonal entries and then take the determanent:A - lambda I = [ 1 - lambda , 2 ; 2 , 1 - lambda ]det ( A - lambda I ) = (1-lambda)(1-lambda) - 4= 1 - 2 lambda + lambda^2 - 4= lambda^2 - 2 lambda - 3This is the "characteristic polynomial"p_A(lambda) = det( A - lambda I )of A , and its roots are the "characteristic values" or "eigenvalues" of A .In our case, p_A(lambda) = (lambda + 1)(lambda - 3)and there are two roots, lambda_1 = -1 and lambda_2 = 3 .[4] Now we can find those special directions. There is one line for lambda_1 and another for lambda_2 . We have to find nonzero solution v to(A - lambda I) v = 0eg with lambda = lambda_1 = -1 , A - lambda = [ 2 2 ; 2 2 ]There is a nontrivial linear relation between the columns:A [ 1 ; -1 ] = 0All we are claiming is thatA [ 1 ; -1 ] = - [ 1 ; -1 ]and you can check this directly. Any such v (even zero) is called an "eigenvector" of A.Back to the differential equation. We have found that there is astraight linesolution of the form r(t) v where v = [1;-1] . We have r' v = u' = A u = A rv = r A v = r lambda vso (since v is nonzero)r' = lambda rand solving this goes straight back to Day One:r = c e^{lambda t}so for us r = c e^{-t} and we have found our first straight line solution:u = e^{-t} [1;-1]In fact we've found all solutions which occur along that line: u = c e^{-t} [1;-1]Any one of these solutions is called a "normal mode."General fact: the eigenvalue turns out to play a much more important rolethan it looked like it would: the straight line solutions are*exponential*solutions, e^{lambda t} v , where lambda is an eigenvalue forthe matrix and v is a nonzero eigenvector for this eigenvalue.The second eigenvalue, lambda_2 = 3 , leads toA - lambda I = [ -1 1 ; 1 -1 ]and [ -1 1 ; 1 -1 ] v = 0 has nonzero solution v = [1;1]so [1;1] is a nonzero eigenvector for the eigenvalue lambda = 3 , and there is another straight line solutione^{3t} [1;1][5] The general solution to u' = Au will be a linear combination of the two eigensolutions (as long as there are two distinct eigenvalues). In our example, the general solution isu = c1 e^{-t} [1 ; -1] + c2 e^{3t} [1 ; 1]We can solve for c1 and c2 using an initial condition: say for exampleu(0) = [2 ; 0]. Well,u(0) = c1 [1 ; -1] + c2 [1 ; 1] = [c1+c2 ; -c1+c2]and for this to be [2 ; 0] we must have c1 = c2 = 1:u(t) = e^{-t} [1 ; -1] + e^{3t} [1 ; 1] .When t is very negative, -10, say, the first term is very big and the second tiny: the solution is very near the line through [1 ; -1].As t gets near zero, the two terms become comparable and the solution curves around. As t gets large, 10, say, the second term is very big and the first is tiny: the solution becomes asymptotic to the line through[1 ; 1].The general solution is a combination of the two normal modes.[6] Comments:(1) The characteristic polynomial for the general 2x2 matrix A = [a,b;c,d] isp_A(lambda) = (a-lambda)(d-lambda) - bc= lambda^2 - (a+d) lambda - (ad-bc)The sum of the diagonal terms of a square matrix is the "trace" of A , tr A,sop_A(lambda) = lambda^2 - (tr A) lambda + (det A)In our example, tr A = 2 and det A = 3 , andp_A(lambda) = lambda^2 - 2 lambda - 3 .(2) Any multiple of an eigenvector is another eigenvector for the same eigenvalue; they form a line, an "eigenline."(3) The eigenlines for distinct eigenvalues are not genearally perpendicularto each other; that is a special feature of *symmetric* matrices, those forwhich b = c .Also, generally the eigenvalues, roots of the characteristic polynomial, may be complex, not real. But for a symmetric matrix, all the eigenvaluesare real.Both these facts hold in higher dimensions as well. Most real numbers we know about are eigenvalues of symmetric matrices - the mass of an elementaryparticle, for example.。

正则化方法赫森矩阵 -回复

正则化方法赫森矩阵-回复什么是正则化方法赫森矩阵？在机器学习和统计学中，正则化是一种常用的方法，用于减小学习算法的复杂度并防止过拟合。

而赫森矩阵（Hessian Matrix）是用于描述多元函数的二阶偏导数的矩阵。

正则化方法和赫森矩阵在机器学习中扮演着重要的角色，通过对样本数据进行适当的惩罚，提高了机器学习算法的泛化能力，并减小了模型的复杂度。

本文将一步一步回答关于正则化方法赫森矩阵的问题，以帮助读者更好地理解它们的原理和应用。

第一部分：正则化方法1. 什么是正则化方法？正则化是一种用于减小模型复杂度并防止过拟合的技术。

它通过在损失函数中增加一个正则化项，对模型的参数进行约束，控制参数的大小，以避免模型在训练数据上过多地关注噪声或异常点，从而提高模型的泛化能力。

2. 为什么需要正则化方法？在机器学习中，模型在训练数据上可能会出现过拟合现象，即模型过于复杂，过度拟合了训练数据中的噪声和异常点，导致在新的数据上表现不佳。

正则化方法通过限制模型的参数大小，降低复杂度，提高了模型在新数据上的预测准确性。

3. 常见的正则化方法有哪些？常见的正则化方法包括L1正则化和L2正则化。

L1正则化通过在损失函数中加入模型参数的绝对值之和，并使得部分参数为0，从而实现特征选择和稀疏性；L2正则化则通过在损失函数中加入模型参数的平方和，并使得参数趋于较小的值，从而平滑模型的参数，并避免过拟合。

第二部分：赫森矩阵1. 什么是赫森矩阵？赫森矩阵是一个描述多元函数的二阶偏导数的矩阵。

对于一个具有n个自变量的函数，赫森矩阵是一个n×n的矩阵，其中每个元素是函数的二阶偏导数。

2. 赫森矩阵有什么作用？赫森矩阵提供了有关函数局部曲率变化的信息，可以帮助我们理解函数在某一点附近的趋势和形状。

尤其在求解优化问题时，赫森矩阵可以帮助我们确定函数的极值点以及该点的性质。

3. 如何计算赫森矩阵？为了计算一个多元函数的赫森矩阵，我们需要求解函数的所有一阶和二阶偏导数。

勤劳强国家的英语作文

Diligence is a cornerstone of success,and it is the key to building a strong nation.A nations prosperity and strength are not built overnight they are the result of the collective efforts of its people.Here is an essay on the importance of diligence in building a strong country.Title:The Role of Diligence in Building a Strong NationIn the tapestry of a nations development,diligence is the golden thread that weaves progress and strength.It is the unwavering commitment to hard work and perseverance that propels a country forward,fostering an environment where innovation,growth,and resilience can thrive.This essay delves into the significance of diligence in the construction of a robust and prosperous nation.Cultivating a Diligent Work EthicThe foundation of a strong nation lies in its people.A diligent work ethic is instilled from an early age,encouraging individuals to take pride in their contributions,no matter how small they may seem.Schools,families,and communities play a pivotal role in nurturing this ethos,teaching the value of time,the satisfaction of a job well done,and the importance of striving for excellence.Economic Growth and DevelopmentDiligence is the driving force behind economic growth.It fuels the engine of industry, where workers dedicate long hours to perfecting their crafts and innovating new solutions. This dedication leads to increased productivity,which in turn attracts investment,creates jobs,and raises the standard of living for all citizens.Innovation and Technological AdvancementA nation that values diligence is a nation that values knowledge and innovation.Diligent individuals are more likely to pursue education and research,pushing the boundaries of science and technology.This leads to breakthroughs that can revolutionize industries, improve public services,and enhance the overall quality of life.Social Cohesion and National IdentityDiligence also plays a crucial role in fostering social cohesion.When citizens see themselves as part of a collective effort to build a better future,it strengthens national identity and unity.This shared sense of purpose can overcome social divides and create amore harmonious society.Overcoming ChallengesNo nation is immune to challenges,whether they be economic downturns,natural disasters,or global crises.A diligent nation,however,is better equipped to face these adversities.The resilience born from a culture of hard work allows a country to bounce back stronger and more prepared for future obstacles.Global CompetitivenessIn an increasingly interconnected world,a nations diligence can be its greatest asset in the global marketplace.By maintaining high standards of work and a relentless pursuit of improvement,a country can outcompete its rivals,secure its place on the world stage,and ensure its longterm survival.ConclusionIn conclusion,diligence is the bedrock upon which a strong nation is built.It is the common thread that runs through economic prosperity,social harmony,and national pride.By fostering a culture of diligence,a country can ensure its continued growth, resilience,and success in an everchanging world.It is through the sweat of our collective brow that we pave the way for future generations to inherit a nation that is not only strong but also just,equitable,and prosperous.。

免疫算法基本流程 -回复

免疫算法基本流程 -回复免疫算法（Immune Algorithm，IA）是仿生学领域的一种元启发式算法，它模仿人类免疫系统的功能，用于解决复杂问题的优化问题。

其基本流程包括问题建模、个体编码、种群初始化、克隆操作、变异操作、选择操作等，接下来本文将从这些方面进一步展开详细描述。

一、问题建模在使用免疫算法解决优化问题之前，需要将问题进行合理的建模。

建模过程主要涉及问题的因素、目标和约束条件等问题，例如在TSP（Traveling Salesman Problem）中，需要定义地图中所有城市之间的距离以及行走路线的长度等因素。

建模完成后，将其转化为适合于免疫算法处理的数学表示形式，这有助于优化算法的精度和效率。

二、个体编码从问题建模后，需要将问题的变量转化为适合免疫算法处理的个体编码，即将问题的解转化成一些序列或数值，这样才能进行算法的操作。

对于不同的问题，需要设计合适的编码方式，例如对于TSP问题，可以将城市序列编码成01字符串等。

三、种群初始化在免疫算法中，需要构建一个种群，种群中的每个个体代表了问题的一个解。

种群初始化是在搜索空间中随机生成一组解，并且保证这些解满足约束条件。

种群大小需要根据问题规模和计算能力来合理安排，一般情况下，种群大小越大，搜索空间越大，但是计算成本也越高。

四、克隆操作在免疫算法中，克隆操作是其中一个重要的基因变异操作。

该操作的目的是产生大量近似于当前最优的个体，增加搜索空间的多样性。

克隆操作的流程如下：1.计算适应度函数值，根据适应度函数值进行排序。

2.选择适应度函数值最优的一部分个体进行克隆操作。

3.对克隆个体进行加密操作，增加其多样性。

5、变异操作变异操作是免疫算法中的一个基本操作，其目的是使部分克隆个体产生和原个体不同的搜索方向，增加搜索空间的变异性。

在变异操作中，采用随机、局部搜索或任意搜索等方法来对某些个体进行改变其参数或某些属性，以期望产生一些新的解。

变异操作的流程如下：1.从克隆群体中随机选择一定数量的个体进行变异操作。

reducedimension method

reducedimension method1. IntroductionThe dimensionality reduction technique known as "Reducedimension" refers to a method used to reduce the number of features in a dataset while preserving the most important information. It is commonly used in machine learning and data analysis tasks to overcome the curse of dimensionality and improve computational efficiency. In this article, we will discuss the principles and procedures of the Reducedimension method.2. Principles of ReducedimensionReducedimension is based on the assumption that the data lies on a low-dimensional manifold embedded in a high-dimensional space. The method aims to find a transformation that maps the original high-dimensional space to a lower-dimensional space while preserving the intrinsic structure and relationships of the data. By reducing the dimensionality, it becomes easier to visualize, analyze, and model the data.3. Procedure of ReducedimensionThe Reducedimension method can be performed in several steps:a. Data preprocessing: Before applying Reducedimension, it is necessary to preprocess the data. This includes handling missing values, normalizing or standardizing the features, and dealing with categorical variables. Data preprocessing ensures that the algorithm performs optimally.b. Covariance matrix computation: The covariance matrix is a symmetric positive semi-definite matrix that represents therelationship between the features in the dataset. It is computed to capture the linear dependencies between the variables.c. Eigenvalue decomposition: By decomposing the covariance matrix, we obtain its eigenvalues and eigenvectors. The eigenvalues represent the amount of variance explained by each eigenvector. The eigenvectors are the directions along which the data varies the most.d. Selection of principal components: The principal components are selected based on the eigenvalues. The eigenvectors corresponding to the largest eigenvalues capture most of the variance in the data. These eigenvectors are chosen as the principal components.e. Projection: The original high-dimensional data is projected onto the newly defined lower-dimensional space spanned by the principal components. The projection reduces the dimensionality of the data while preserving its essential properties and minimizing information loss.f. Reconstruction: If needed, the reduced-dimensional data can be reconstructed back into the original high-dimensional space using the inverse projection matrix. This allows for analysis or visualization in the original feature space.4. Advantages of ReducedimensionThe Reducedimension method offers several advantages:a. Dimensionality reduction: The method reduces thedimensionality of the dataset, which helps to overcome the curse of dimensionality. It simplifies the data representation and improves computational efficiency.b. Feature selection: Reducedimension automatically selects the most informative features by calculating the eigenvalues and eigenvectors. It eliminates redundant or irrelevant features, resulting in a more concise and interpretable dataset.c. Intrinsic structure preservation: Reducedimension aims to preserve the intrinsic structure and relationships of the data during the dimensionality reduction process. It ensures that the important information is retained while discarding noise and irrelevant variations.5. Applications of ReducedimensionThe Reducedimension method has various applications in various fields, including:a. Image and signal processing: It is used to reduce the dimensionality of image and signal data, enabling efficient compression, denoising, and feature extraction.b. Pattern recognition: Reducedimension is applied to extract discriminative features from high-dimensional datasets, improving the performance of pattern recognition algorithms.c. Bioinformatics: It is used to analyze and visualize genomic and proteomic data, enabling the identification of key genes or proteins associated with specific diseases or biological processes.d. Financial analysis: Reducedimension is used to analyze and model financial data, identifying the key factors that drive stock prices or predicting market trends.6. ConclusionThe Reducedimension method provides an effective approach for reducing the dimensionality of high-dimensional datasets while preserving the most relevant information. Its principles and procedures, including data preprocessing, covariance matrix computation, eigenvalue decomposition, principal component selection, projection, and reconstruction, enable efficient analysis, modeling, and visualization of complex datasets in various fields.。

A tutorial on Principal Components Analysis

A tutorial on Principal Components AnalysisLindsay I SmithFebruary26,2002Chapter1IntroductionThis tutorial is designed to give the reader an understanding of Principal Components Analysis(PCA).PCA is a useful statistical technique that has found application in ﬁelds such as face recognition and image compression,and is a common technique for ﬁnding patterns in data of high dimension.Before getting to a description of PCA,this tutorialﬁrst introduces mathematical concepts that will be used in PCA.It covers standard deviation,covariance,eigenvec-tors and eigenvalues.This background knowledge is meant to make the PCA section very straightforward,but can be skipped if the concepts are already familiar.There are examples all the way through this tutorial that are meant to illustrate the concepts being discussed.If further information is required,the mathematics textbook “Elementary Linear Algebra5e”by Howard Anton,Publisher John Wiley&Sons Inc, ISBN0-471-85223-6is a good source of information regarding the mathematical back-ground.1Chapter2Background MathematicsThis section will attempt to give some elementary background mathematical skills that will be required to understand the process of Principal Components Analysis.The topics are covered independently of each other,and examples given.It is less important to remember the exact mechanics of a mathematical technique than it is to understand the reason why such a technique may be used,and what the result of the operation tells us about our data.Not all of these techniques are used in PCA,but the ones that are not explicitly required do provide the grounding on which the most important techniques are based.I have included a section on Statistics which looks at distribution measurements, or,how the data is spread out.The other section is on Matrix Algebra and looks at eigenvectors and eigenvalues,important properties of matrices that are fundamental to PCA.2.1StatisticsThe entire subject of statistics is based around the idea that you have this big set of data, and you want to analyse that set in terms of the relationships between the individual points in that data set.I am going to look at a few of the measures you can do on a set of data,and what they tell you about the data itself.2.1.1Standard DeviationTo understand standard deviation,we need a data set.Statisticians are usually con-cerned with taking a sample of a population.To use election polls as an example,the population is all the people in the country,whereas a sample is a subset of the pop-ulation that the statisticians measure.The great thing about statistics is that by only measuring(in this case by doing a phone survey or similar)a sample of the population, you can work out what is most likely to be the measurement if you used the entire pop-ulation.In this statistics section,I am going to assume that our data sets are samples2of some bigger population.There is a reference later in this section pointing to more information about samples and populations.Here’s an example set:I could simply use the symbol to refer to this entire set of numbers.If I want to refer to an individual number in this data set,I will use subscripts on the symbol to indicate a speciﬁc number.Eg.refers to the3rd number in,namely the number4.Note that is theﬁrst number in the sequence,not like you may see in some textbooks.Also,the symbol will be used to refer to the number of elements in the setThere are a number of things that we can calculate about a data set.For example, we can calculate the mean of the sample.I assume that the reader understands what the mean of a sample is,and will only give the formula:Set1:Total208Square Root8.32668-249-1111111224Divided by(n-1) 3.333Table2.1:Calculation of standard deviationdifference between each of the denominators.It also discusses the difference between samples and populations.So,for our two data sets above,the calculations of standard deviation are in Ta-ble2.1.And so,as expected,theﬁrst set has a much larger standard deviation due to the fact that the data is much more spread out from the mean.Just as another example,the data set:also has a mean of10,but its standard deviation is0,because all the numbers are the same.None of them deviate from the mean.2.1.2VarianceVariance is another measure of the spread of data in a data set.In fact it is almost identical to the standard deviation.The formula is this:You will notice that this is simply the standard deviation squared,in both the symbol ()and the formula(there is no square root in the formula for variance).is the usual symbol for variance of a sample.Both these measurements are measures of the spread of the data.Standard deviation is the most common measure,but variance is also used.The reason why I have introduced variance in addition to standard deviation is to provide a solid platform from which the next section,covariance,can launch from. ExercisesFind the mean,standard deviation,and variance for each of these data sets.[12233444597098][12152527328899][15357882909597]协方差2.1.3CovarianceThe last two measures we have looked at are purely1-dimensional.Data sets like this could be:heights of all the people in the room,marks for the last COMP101exam etc. However many data sets have more than one dimension,and the aim of the statistical analysis of these data sets is usually to see if there is any relationship between the dimensions.For example,we might have as our data set both the height of all the students in a class,and the mark they received for that paper.We could then perform statistical analysis to see if the height of a student has any effect on their mark.Standard deviation and variance only operate on1dimension,so that you could only calculate the standard deviation for each dimension of the data set independently of the other dimensions.However,it is useful to have a similar measure toﬁnd out how much the dimensions vary from the mean with respect to each other.Covariance is such a measure.Covariance is always measured between2dimen-sions.If you calculate the covariance between one dimension and itself,you get the variance.So,if you had a3-dimensional data set(,,),then you could measure the covariance between the and dimensions,the and dimensions,and the and dimensions.Measuring the covariance between and,or and,or and would give you the variance of the,and dimensions respectively.The formula for covariance is very similar to the formula for variance.The formula for variance could also be written like this:5includegraphicscovPlot.psFigure2.1:A plot of the covariance data showing positive relationship between the number of hours studied against the mark receivedIt is exactly the same except that in the second set of brackets,the’s are replaced by ’s.This says,in English,“For each data item,multiply the difference between the value and the mean of,by the the difference between the value and the mean of. Add all these up,and divide by”.How does this work?Lets use some example data.Imagine we have gone into the world and collected some2-dimensional data,say,we have asked a bunch of students how many hours in total that they spent studying COSC241,and the mark that they received.So we have two dimensions,theﬁrst is the dimension,the hours studied, and the second is the dimension,the mark received.Figure2.2holds my imaginarydata,and the calculation of,the covariance between the Hours of study done and the Mark received.So what does it tell us?The exact value is not as important as it’s sign(ie.positive or negative).If the value is positive,as it is here,then that indicates that both di-mensions increase together,meaning that,in general,as the number of hours of study increased,so did theﬁnal mark.If the value is negative,then as one dimension increases,the other decreases.If we had ended up with a negative covariance here,then that would have said the opposite, that as the number of hours of study increased the theﬁnal mark decreased.In the last case,if the covariance is zero,it indicates that the two dimensions are independent of each other.The result that mark given increases as the number of hours studied increases can be easily seen by drawing a graph of the data,as in Figure2.1.3.However,the luxury of being able to visualize data is only available at2and3dimensions.Since the co-variance value can be calculated between any2dimensions in a data set,this technique is often used toﬁnd relationships between dimensions in high-dimensional data sets where visualisation is difﬁcult.You might ask“is equal to”?Well,a quick look at the for-mula for covariance tells us that yes,they are exactly the same since the only dif-ference between and is that is replaced by .And since multiplication is commutative,which means that it doesn’t matter which way around I multiply two numbers,I always get the same num-ber,these two equations give the same answer.2.1.4The covariance MatrixRecall that covariance is always measured between2dimensions.If we have a data set with more than2dimensions,there is more than one covariance measurement that can be calculated.For example,from a3dimensional data set(dimensions,,)you could calculate,,and.In fact,for an-dimensional data set,you can calculateHours(H)Mark(M)Totals167749 Covariance:939-23.421.08-6.93259330.580.08-0.111050-12.424.0851.33032-30.422.0846.97542-20.425.0838.511666 3.586.08106.891149.89104.54 Table2.2:2-dimensional data set and covariance calculation7A useful way to get all the possible covariance values between all the different dimensions is to calculate them all and put them in a matrix.I assume in this tutorial that you are familiar with matrices,and how they can be deﬁned.So,the deﬁnition for the covariance matrix for a set of data with dimensions is:where is a matrix with rows and columns,and is the th dimension. All that this ugly looking formula says is that if you have an-dimensional data set, then the matrix has rows and columns(so is square)and each entry in the matrix is the result of calculating the covariance between two separate dimensions.Eg.the entry on row2,column3,is the covariance value calculated between the2nd dimension and the3rd dimension.An example.We’ll make up the covariance matrix for an imaginary3dimensional data set,using the usual dimensions,and.Then,the covariance matrix has3rows and3columns,and the values are this:Some points to note:Down the main diagonal,you see that the covariance value is between one of the dimensions and itself.These are the variances for that dimension. The other point is that since,the matrix is symmetrical about the main diagonal.ExercisesWork out the covariance between the and dimensions in the following2dimen-sional data set,and describe what the result indicates about the data.Item Number:243923433220131411-1Figure2.2:Example of one non-eigenvector and one eigenvectorFigure2.3:Example of how a scaled eigenvector is still and eigenvector2.2.1EigenvectorsAs you know,you can multiply two matrices together,provided they are compatible sizes.Eigenvectors are a special case of this.Consider the two multiplications between a matrix and a vector in Figure2.2.In theﬁrst example,the resulting vector is not an integer multiple of the original vector,whereas in the second example,the example is exactly4times the vector we began with.Why is this?Well,the vector is a vector in2dimensional space.The vector(from the second example multiplication)represents an arrow pointing from the origin,,to the point.The other matrix,the square one,can be thought of as a transformation matrix.If you multiply this matrix on the left of a vector,the answer is another vector that is transformed from it’s original position.It is the nature of the transformation that the eigenvectors arise from.Imagine a transformation matrix that,when multiplied on the left,reﬂected vectors in the line .Then you can see that if there were a vector that lay on the line,it’s reﬂection it itself.This vector(and all multiples of it,because it wouldn’t matter howlong the vector was),would be an eigenvector of that transformation matrix.What properties do these eigenvectors have?You shouldﬁrst know that eigenvec-tors can only be found for square matrices.And,not every square matrix has eigen-vectors.And,given an matrix that does have eigenvectors,there are of them. Given a matrix,there are3eigenvectors.Another property of eigenvectors is that even if I scale the vector by some amount before I multiply it,I still get the same multiple of it as a result,as in Figure2.3.This is because if you scale a vector by some amount,all you are doing is making it longer,9not changing it’s stly,all the eigenvectors of a matrix are perpendicular,ie.at right angles to each other,no matter how many dimensions you have.By the way, another word for perpendicular,in maths talk,is orthogonal.This is important becauseit means that you can express the data in terms of these perpendicular eigenvectors,instead of expressing them in terms of the and axes.We will be doing this later in the section on PCA.Another important thing to know is that when mathematiciansﬁnd eigenvectors, they like toﬁnd the eigenvectors whose length is exactly one.This is because,as you know,the length of a vector doesn’t affect whether it’s an eigenvector or not,whereas the direction does.So,in order to keep eigenvectors standard,whenever weﬁnd an eigenvector we usually scale it to make it have a length of1,so that all eigenvectors have the same length.Here’s a demonstration from our example above.is an eigenvector,and the length of that vector isso we divide the original vector by this much to make it have a length of1.ExercisesFor the following square matrix:Decide which,if any,of the following vectors are eigenvectors of that matrix and give the corresponding eigenvalue.11Chapter3Principal Components Analysis Finally we come to Principal Components Analysis(PCA).What is it?It is a way of identifying patterns in data,and expressing the data in such a way as to highlight their similarities and differences.Since patterns in data can be hard toﬁnd in data of high dimension,where the luxury of graphical representation is not available,PCA is a powerful tool for analysing data.The other main advantage of PCA is that once you have found these patterns in the data,and you compress the data,ie.by reducing the number of dimensions,without much loss of information.This technique used in image compression,as we will see in a later section.This chapter will take you through the steps you needed to perform a Principal Components Analysis on a set of data.I am not going to describe exactly why the technique works,but I will try to provide an explanation of what is happening at each point so that you can make informed decisions when you try to use this technique yourself.3.1MethodStep1:Get some dataIn my simple example,I am going to use my own made-up data set.It’s only got2 dimensions,and the reason why I have chosen this is so that I can provide plots of the data to show what the PCA analysis is doing at each step.The data I have used is found in Figure3.1,along with a plot of that data.Step2:Subtract the meanFor PCA to work properly,you have to subtract the mean from each of the data dimen-sions.The mean subtracted is the average across each dimension.So,all the values have(the mean of the values of all the data points)subtracted,and all the values have subtracted from them.This produces a data set whose mean is zero.12Data =x2.50.72.22.23.12.721.11.50.9DataAdjust =x .69-1.21.39.291.29.79.19-.81-.31-1.01-101234-101234Original PCA data"./PCAdata.dat"Figure 3.1:PCA example data,original data on the left,data with the means subtracted on the right,and a plot of the data13Step3:Calculate the covariance matrixThis is done in exactly the same way as was discussed in section2.1.4.Since the data is2dimensional,the covariance matrix will be.There are no surprises here,so I will just give you the result:So,since the non-diagonal elements in this covariance matrix are positive,we should expect that both the and variable increase together.Step4:Calculate the eigenvectors and eigenvalues of the covariance matrixSince the covariance matrix is square,we can calculate the eigenvectors and eigenval-ues for this matrix.These are rather important,as they tell us useful information about our data.I will show you why soon.In the meantime,here are the eigenvectors and eigenvalues:It is important to notice that these eigenvectors are both unit eigenvectors ie.their lengths are both1.This is very important for PCA,but luckily,most maths packages, when asked for eigenvectors,will give you unit eigenvectors.So what do they mean?If you look at the plot of the data in Figure3.2then you can see how the data has quite a strong pattern.As expected from the covariance matrix, they two variables do indeed increase together.On top of the data I have plotted both the eigenvectors as well.They appear as diagonal dotted lines on the plot.As stated in the eigenvector section,they are perpendicular to each other.But,more importantly, they provide us with information about the patterns in the data.See how one of the eigenvectors goes through the middle of the points,like drawing a line of bestﬁt?That eigenvector is showing us how these two data sets are related along that line.The second eigenvector gives us the other,less important,pattern in the data,that all the points follow the main line,but are off to the side of the main line by some amount.So,by this process of taking the eigenvectors of the covariance matrix,we have been able to extract lines that characterise the data.The rest of the steps involve trans-forming the data so that it is expressed in terms of them lines.Step5:Choosing components and forming a feature vectorHere is where the notion of data compression and reduced dimensionality comes into it.If you look at the eigenvectors and eigenvalues from the previous section,you14-2-1.5-1-0.50.511.52-2-1.5-1-0.500.51 1.52Mean adjusted data with eigenvectors overlayed"PCAdataadjust.dat"(-.740682469/.671855252)*x (-.671855252/-.740682469)*x Figure 3.2:A plot of the normalised data (mean subtracted)with the eigenvectors of the covariance matrix overlayed on top.15will notice that the eigenvalues are quite different values.In fact,it turns out that the eigenvector with the highest eigenvalue is the principle component of the data set. In our example,the eigenvector with the larges eigenvalue was the one that pointed down the middle of the data.It is the most signiﬁcant relationship between the data dimensions.In general,once eigenvectors are found from the covariance matrix,the next step is to order them by eigenvalue,highest to lowest.This gives you the components in order of signiﬁcance.Now,if you like,you can decide to ignore the components of lesser signiﬁcance.You do lose some information,but if the eigenvalues are small,you don’t lose much.If you leave out some components,theﬁnal data set will have less dimensions than the original.To be precise,if you originally have dimensions in your data,and so you calculate eigenvectors and eigenvalues,and then you choose only theﬁrst eigenvectors,then theﬁnal data set has only dimensions.What needs to be done now is you need to form a feature vector,which is just a fancy name for a matrix of vectors.This is constructed by taking the eigenvectors that you want to keep from the list of eigenvectors,and forming a matrix with these eigenvectors in the columns.Given our example set of data,and the fact that we have2eigenvectors,we have two choices.We can either form a feature vector with both of the eigenvectors:or,we can choose to leave out the smaller,less signiﬁcant component and only have a single column:We shall see the result of each of these in the next section.Step5:Deriving the new data setThis theﬁnal step in PCA,and is also the easiest.Once we have chosen the components (eigenvectors)that we wish to keep in our data and formed a feature vector,we simply take the transpose of the vector and multiply it on the left of the original data set, transposed.where is the matrix with the eigenvectors in the columns trans-posed so that the eigenvectors are now in the rows,with the most signiﬁcant eigenvec-tor at the top,and is the mean-adjusted data transposed,ie.the data items are in each column,with each row holding a separate dimension.I’m sorry if this sudden transpose of all our data confuses you,but the equations from here on are16easier if we take the transpose of the feature vector and the dataﬁrst,rather that having a little T symbol above their names from now on.is theﬁnal data set,with data items in columns,and dimensions along rows.What will this give us?It will give us the original data solely in terms of the vectors we chose.Our original data set had two axes,and,so our data was in terms of them.It is possible to express data in terms of any two axes that you like.If these axes are perpendicular,then the expression is the most efﬁcient.This was why it was important that eigenvectors are always perpendicular to each other.We have changed our data from being in terms of the axes and,and now they are in terms of our2 eigenvectors.In the case of when the new data set has reduced dimensionality,ie.we have left some of the eigenvectors out,the new data is only in terms of the vectors that we decided to keep.To show this on our data,I have done theﬁnal transformation with each of the possible feature vectors.I have taken the transpose of the result in each case to bring the data back to the nice table-like format.I have also plotted theﬁnal points to show how they relate to the components.In the case of keeping both eigenvectors for the transformation,we get the data and the plot found in Figure3.3.This plot is basically the original data,rotated so that the eigenvectors are the axes.This is understandable since we have lost no information in this decomposition.The other transformation we can make is by taking only the eigenvector with the largest eigenvalue.The table of data resulting from that is found in Figure3.4.As expected,it only has a single dimension.If you compare this data set with the one resulting from using both eigenvectors,you will notice that this data set is exactly the ﬁrst column of the other.So,if you were to plot this data,it would be1dimensional, and would be points on a line in exactly the positions of the points in the plot in Figure3.3.We have effectively thrown away the whole other axis,which is the other eigenvector.So what have we done here?Basically we have transformed our data so that is expressed in terms of the patterns between them,where the patterns are the lines that most closely describe the relationships between the data.This is helpful because we have now classiﬁed our data point as a combination of the contributions from each of those lines.Initially we had the simple and axes.This isﬁne,but the and values of each data point don’t really tell us exactly how that point relates to the rest of the data.Now,the values of the data points tell us exactly where(ie.above/below)the trend lines the data point sits.In the case of the transformation using both eigenvectors, we have simply altered the data so that it is in terms of those eigenvectors instead of the usual axes.But the single-eigenvector decomposition has removed the contribution due to the smaller eigenvector and left us with data that is only in terms of the other.3.1.1Getting the old data backWanting to get the original data back is obviously of great concern if you are using the PCA transform for data compression(an example of which to will see in the next section).This content is taken fromhttp://www.vision.auc.dk/sig/Teaching/Flerdim/Current/hotelling/hotelling.html17Transformed Data=-.827970186.142857227-.992197494.130417207-1.67580142.175282444.0991094375.0464172582.438046137-.162675287-2-1.5-1-0.50.511.52-2-1.5-1-0.500.51 1.52Data transformed with 2 eigenvectors"./doublevecfinal.dat"Figure 3.3:The table of data by applying the PCA analysis using both eigenvectors,and a plot of the new data points.18Transformed Data(Single eigenvector)-101234-101234Original data restored using only a single eigenvector"./lossyplusmean.dat"Figure 3.5:The reconstruction from the data that was derived using only a single eigen-vectorit to the original data plot in Figure 3.1and you will notice how,while the variationalong the principle eigenvector (see Figure 3.2for the eigenvector overlayed on top ofthe mean-adjusted data)has been kept,the variation along the other component (theother eigenvector that we left out)has gone.ExercisesWhat do the eigenvectors of the covariance matrix give us?At what point in the PCA process can we decide to compress the data?Whateffect does this have?For an example of PCA and a graphical representation of the principal eigenvec-tors,research the topic ’Eigenfaces’,which uses PCA to do facial recognition20Chapter4Application to Computer VisionThis chapter will outline the way that PCA is used in computer vision,ﬁrst showing how images are usually represented,and then showing what PCA can allow us to dowith those images.The information in this section regarding facial recognition comes from“Face Recognition:Eigenface,Elastic Matching,and Neural Nets”,Jun Zhang etal.Proceedings of the IEEE,V ol.85,No.9,September1997.The representation infor-mation,is taken from“Digital Image Processing”Rafael C.Gonzalez and Paul Wintz, Addison-Wesley Publishing Company,1987.It is also an excellent reference for further information on the K-L transform in general.The image compression information is taken from http://www.vision.auc.dk/sig/Teaching/Flerdim/Current/hotelling/hotelling.html, which also provides examples of image reconstruction using a varying amount of eigen-vectors.4.1RepresentationWhen using these sort of matrix techniques in computer vision,we must consider repre-sentation of images.A square,by image can be expressed as an-dimensional vectorwhere the rows of pixels in the image are placed one after the other to form a one-dimensional image.E.g.Theﬁrst elements(will be theﬁrst row of the image,the next elements are the next row,and so on.The values in the vector arethe intensity values of the image,possibly a single greyscale value.4.2PCA toﬁnd patternsSay we have20images.Each image is pixels high by pixels wide.For each image we can create an image vector as described in the representation section.Wecan then put all the images together in one big image-matrix like this:21which gives us a starting point for our PCA analysis.Once we have performed PCA, we have our original data in terms of the eigenvectors we found from the covariance matrix.Why is this useful?Say we want to do facial recognition,and so our original images were of peoples faces.Then,the problem is,given a new image,whose face from the original set is it?(Note that the new image is not one of the20we started with.)The way this is done is computer vision is to measure the difference between the new image and the original images,but not along the original axes,along the new axes derived from the PCA analysis.It turns out that these axes works much better for recognising faces,because the PCA analysis has given us the original images in terms of the differences and simi-larities between them.The PCA analysis has identiﬁed the statistical patterns in the data.Since all the vectors are dimensional,we will get eigenvectors.In practice, we are able to leave out some of the less signiﬁcant eigenvectors,and the recognition still performs well.4.3PCA for image compressionUsing PCA for image compression also know as the Hotelling,or Karhunen and Leove (KL),transform.If we have20images,each with pixels,we can form vectors, each with20dimensions.Each vector consists of all the intensity values from the same pixel from each picture.This is different from the previous example because before we had a vector for image,and each item in that vector was a different pixel,whereas now we have a vector for each pixel,and each item in the vector is from a different image.Now we perform the PCA on this set of data.We will get20eigenvectors because each vector is20-dimensional.To compress the data,we can then choose to transform the data only using,say15of the eigenvectors.This gives us aﬁnal data set with only15dimensions,which has saved us of the space.However,when the original data is reproduced,the images have lost some of the information.This compression technique is said to be lossy because the decompressed image is not exactly the same as the original,generally worse.22。

全模型迭代重建在头颈部CT血管成像中的应用

国际医学放射学杂志ＩｎｔＪＭｅｄＲａｄｉｏｌ２０19July 鸦42穴4雪：418-421全模型迭代重建在头颈部CT 血管成像中的应用吴晓玲黄飚*【摘要】全模型迭代重建（IMR ）在重建过程中建立图像统计模型和数据统计模型，通过反复进行扫描模型与采集数据之间的差异对比校正，实现影像最优化。

头颈部CT 血管成像采用IMR 技术能显著降低影像噪声、提高影像信噪比和对比噪声比，可增强对脑动脉硬化和动脉瘤的诊断信心；还能提高对早期缺血性卒中动脉致密征检出的敏感度、大幅降低CT 灌注的辐射剂量、减少CT 血管成像对比剂用量以及降低病人的碘负荷。

综述IMR 技术在头颈部CT 血管成像中的应用进展。

【关键词】全模型迭代重建技术；CT 血管成像；辐射剂量中图分类号：R445.3；R814.42文献标志码：AApplication of knowledge-based iterative model reconstruction in head-neck CT angiography WU Xiaoling,HUANG Biao.Department of Radiology,Guangdong General Hospital,School of Medicine,South China University of Technology,Guangzhou 510080,China【Abstract 】Iterative model reconstruction (IMR)establishes image statistics and data statistics models during itsiterative cycle and achieves image optimization by repeatedly correcting the differences between scan model and acquired data.IMR can significantly reduce image noise,improve image signal-to-noise ratio and contrast-to-noise ratio for head and neck CTA,and increase the diagnostic confidence of atherosclerosis and aneurysm.IMR can also increase sensitivity in detecting hyperdense artery sign in the early ischemic stroke,significantly reduce the radiation dose in CT perfusion and the iodine load of patients.In this paper,the progress of application of iterative model reconstruction technology in head and neck CT angiography was reviewed.【Keywords 】Iterative model reconstruction;CT angiography;Radiation doseＩｎｔＪＭｅｄＲａｄｉｏｌ，２０１9，42（4）：418-421作者单位：华南理工大学附属广东省人民医院放射科，广州510080通信作者：黄飚，E-mail:cjr.huangbiao@ *审校者DOI:10.19300/j.2019.Z6654CT 血管成像（CT angiography ，CTA ）具有无创、简便、快速等优点，对于脑血管疾病的诊断和评估具有重要价值。

应用荧光实时定量PCR方法检测重组慢病毒滴度及其感染效率（精品）

生命科学研究2009年慢病毒载体是目前应用最广泛的基因运载工具之一，在基因治疗研究和转基因动物的制备中已显示出其广阔的应用前景．慢病毒载体是一种反转录病毒载体，其中以人类免疫缺陷性病毒（HIV -1）载体研究最为深入．与传统的小鼠白血病病毒载体（MuLV ）偏爱整合入基因5′端相比，慢病毒载体均匀分布于基因组内，从而降低了其激活原癌基因的几率［1~3］．重组慢病毒载体既能感染分裂期细胞，也能感染非分裂期细胞［4］．它可携带较大的外源基因（约8kb 左右）并稳定整合和表达［5］，加之其诱发的宿主免疫反应相对较小［6］，使得重组慢病毒载体具有较为广阔的应用应用荧光实时定量PCR 方法检测重组慢病毒滴度及其感染效率马海燕，方彧聃，张敬之*（上海交通大学医学院，上海市儿童医院上海市医学遗传研究所，中国上海200040）摘要：慢病毒载体已经广泛应用于动物模型中基因治疗的研究和转基因动物的制备，而准确地测定重组慢病毒的滴度和感染效率是其关键步骤．通过荧光实时定量PCR 的方法定量分析重组慢病毒的颗粒数以及病毒的活性滴度，并以GFP 报告基因的方法作为对照来验证定量PCR 方法的准确性．研究结果显示，应用荧光实时定量PCR 法与GFP 报告基因法测定得到的病毒活性滴度成正相关，而且前者可以更加准确地测定病毒滴度和病毒感染效率．关键词：慢病毒载体；荧光实时定量PCR ；病毒滴度；整合拷贝数；感染效率中图分类号：Q331文献标识码：A文章编号：1007-7847（2009）05-0394-05A Novel Method for the Determination of Recombinant LentiviralTiter and Infectivity by qRT -PCRMA Hai -yan ，FANG Yu -dan ，ZHANG Jing -zhi *（School of Medicine ，Shanghai Jiaotong University ，Shanghai Children ’s Hospital ，Shanghai Institute of Medical Genetics ，Shanghai 200040，China ）收稿日期：2009-03-18；修回日期：2009-05-16基金项目：国家高技术研究发展计划项目（2007AA021206）；国家自然科学基金资助项目（30870943）；上海市自然科学基金资助项目（08ZR1412100）作者简介：马海燕（1983-），女，山东淄博人，硕士研究生，主要从事病毒载体在转基因动物制备中的应用研究；*通讯作者：张敬之（1959-），男，上海人，上海交通大学医学遗传研究所副教授，博士，主要从事分子病毒学研究，Tel ：021-********，E -mail ：********************.Abstract ：Lentiviral vector is being widely used in the study of gene therapy in animal models and in generating transgenic animals.However ，determination of lentiviral particles and their infectivity is essential before their being used.Such a requirement can be accurately achieved by qRT -PCR.Refered by infectious units got from GFP reporter assay ，it showed a positive correlation between the two approaches.A reliable ，accurate and rapid method is therefore established for the determination of the recombinant lentiviral titer and the infectivity.Key words ：lentiviral vector ；qRT -PCR ；viral titer ；integration copy number ；infectivity（Life Science Research ，2009，13（5）：394～398）第13卷第5期生命科学研究Vol．13No．52009年10月Life Science Research Oct.2009第5期前景．无论何种目的使用重组慢病毒，都有必要准确地检测病毒滴度．目前，重组慢病毒滴度的检测方法有：依赖于报告基因的GFP荧光检测法、检测慢病毒外壳蛋白p24的抗原-抗体法（ELISA 法）和检测其逆转录酶活性的酶学法等．这些方法多存在耗时、费力、检测成本较高、病毒用量多、不适于非报告基因载体等缺点．所以，建立一种快速、简便、准确的慢病毒滴度检测方法，是非常有必要的．在此，我们介绍一种利用实时定量PCR测定重组慢病毒滴度的方法．该方法通过在载体的长末端重复序列区（LTR）设计定量引物，利用荧光实时定量PCR测定重组慢病毒中LTR拷贝数来测定病毒颗粒数和有效感染的病毒颗粒数．通过与GFP报告基因测定法的比较验证，证明该测定方法准确可靠．并且，通过慢病毒颗粒数与实际有活力病毒滴度的比较，可以计算得到重组慢病毒感染效率．多次重复实验证明，该方法具有快速、准确的优点，非常适用于非报告基因载体的病毒滴度及其感染效率的测定．1材料与方法1.1材料组成慢病毒载体的3个质粒FUGW（即含eGFP基因的转基因质粒），△8.9（编码结构和非结构蛋白基因质粒）和VSVG（外壳蛋白质粒）由美国Marine Medical Center Research Institute王征宇博士惠赠；293T细胞（人胚肾细胞）购自美国ATCC（American Type Culture Collection）细胞库；ProFection购自Promega公司；胎牛血清（FBS）、培养液和Hanks液及Salmon Sperm DNA 购自Gibco BRL；STE配方为10mmol/L Tris pH 7.6，1mmol/L EDTA pH8.0，0.1mol/L NaCl．1.2FUGW病毒的制备当293T细胞长至70%饱和度时，用磷酸钙法转染，按照Promega公司的ProFection Kit说明书操作，其中FUGW15μg，△8.910μg，VSVG7.5μg，转染6h后换完全培养液（含10% FBS的DMEM），并在37℃，5%CO2培养约60h．上述病毒培养上清液经离心、过滤后，50000g 超速离心1.5h后弃上清，在病毒沉淀上加少量Hanks液，获得病毒浓缩液，-80℃保存待用．1.3病毒RNA的提取取2μL病毒浓缩液进行DNase处理，体系中加入5μL10×DNase I Buffer、2μL DNase I （5U/uL，Takara Bio Inc，Shiga，Japan）、2μL RNase inhabitor（Takara Bio Inc.，Shiga，Japan），用DEPC水定容至50μL反应体系，37℃45 min．DNase处理后的混合液，加入350μL STE、20μL10%SDS和5μL蛋白酶K（20g/L，AMRESCO，Solon，OH），56℃15min水解．最后等体积酚、氯仿抽提，两倍体积无水乙醇沉淀，冻干后，20μL DEPC水溶解．1.4反转录反应病毒RNA在酚/氯仿抽提、无水乙醇沉淀前，需经过DNase I处理，以避免DNA污染．根据产品说明书，取1μg RNA、20pmol RT-PCR 下游引物（5′-GAGAGCTCCCAGGCTCAGATC-3′）、2μL5×RT Buffer、1μL MLV酶（Takara Bio Inc，Shiga，Japan）、水补足至10μL体系，37℃1h．1.5实时定量PCR分析病毒颗粒数为了测定制备的慢病毒的病毒颗粒数，应用实时定量PCR测定病毒LTR拷贝数.其中，引物和探针序列见表1．在反应体系中，引物900 nmol/L，探针250nmol/L，2.5μL10×Ex-Buffer，15μmol Mg2+，2.5μmol dNTP，1U ExTaqE，5μL样本，总反应体系为25μL体积．实时定量PCR仪（Corbett Life Science RG-3000，Sidney，Australia）上反应：95℃5min变性，95℃30s，59℃30s，40个循环，59℃520nm处检测荧光值．软件分析荧光检测数据．1.6不同剂量FUGW病毒感染293T细胞取病毒浓缩液，按10倍稀释法，取0.1、0.01、0.001μL病毒浓缩液（即用实时定量PCR 检测法，定量慢病毒颗粒数为：6.32×107、6.32×106、6.32×105），用含有8mg/L Polybrene促感染（Sigma-Aldrich，Inc，St.Louis，MO）且不含血清的培养液逐级稀释后，感染293T细胞（1.5×106/孔），37℃2h.然后，加入完全培养液培养细胞2d，表1实时定量PCR引物及探针序列Table1The sequences of primer and probe of Real-time PCRLTR-FLTR-PLTR-ProbePrimer Sequences5′-ACAGCCGCCTAGCATTTCAT-3′5′-GAGAGCTCCCAGGCTCAGATC-3′5′-ACATGGCCCGAGAGCTGCATCC-3′马海燕等：应用荧光实时定量PCR方法检测重组慢病毒滴度及其感染效率395生命科学研究2009年图1定量PCR 反应荧光强度曲线Fig.1Fluorescence intensity curve of Real -time PCR图2定量PCR 标准曲线Fig.2Standard curve of Real -time PCR至荧光显微镜下观测绿色荧光蛋白表达情况．1.7病毒感染细胞内DNA 的提取细胞经2d 培养后，用胰酶将细胞消化，收集入1.5mL EP tube 中，200g ，5min ，将细胞沉淀下来．加入200μL STE 、20μL 10%SDS 、10μL 蛋白酶K ，混匀后，37℃4h.加入等体积酚，振荡混匀，15000g ，离心12min ；吸取上清，加入等体积的氯仿，振荡混匀，15000g 离心6min ；吸取上清，加入1/10体积3mol/mL NaAc ，两倍体积的无水乙醇，-20℃沉淀1h 以上；混合液15000g ，4℃离心20min ；弃上清，沉淀风干，100μL TE 溶解．1.8实时定量PCR 测定重组载体整合拷贝数为了测定被感染的293T 细胞内重组载体的整合，用实时定量PCR 检测上述被抽提的基因组DNA 中LTR 的拷贝数．具体方法同1.5．2结果2.1实时定量PCR 检测病毒颗粒数本实验中，定量PCR 引物设计在LTR 区，由于一个慢病毒含有两个病毒基因拷贝，因此，在计算病毒颗粒数时，LTR 拷贝数除以2即为病毒颗粒数.应用实时定量PCR 检测到病毒LTR 拷贝数为1.26×1012/mL ，定量PCR 反应荧光强度曲线及标准曲线见图1、图2.通过计算，得到病毒颗粒数为6.32×1011/mL ．此时计算得到的病毒颗粒数为所有收集到的病毒颗粒总数，既包括有感染效力的病毒，也包括无感染效力的病毒．为了测定制备得到的病毒实际滴度（即单位体积内有感染效力的病毒颗粒数），我们将不同剂量病毒感染细胞，分别用GFP 报告基因法和定量PCR 法测定病毒滴度．2.2实时定量PCR 检测病毒载体整合拷贝数，计算病毒滴度取病毒浓缩液，按照10倍稀释，分别取0.1、0.01、0.001μL 病毒浓缩液感染1.5×106293T 细胞，感染2d 后，应用实时定量PCR 检测不同病毒量感染的细胞DNA 中LTR 整合拷贝数，结果显示，0.1、0.01、0.001μL 病毒感染的细胞中外源基因整合的拷贝数分别为5.32×106、9.28×105、4.48×104.定量PCR 的系统参数为：R =0.99，R ∧2=0.99，Efficiency =1.01，各参数值表明实验检测的准确性和可信性．当一个病毒感染细胞并将外源基因整合入基因组后，由于其末端发生跳跃过程，使得每条DNA 单链上含有两个完整的LTR ，因此，在被整合的细胞基因组，一个慢病毒颗粒=LTR 拷贝数/4．通过计算，得到病毒滴度为（1.59±0.64）×1010IU /mL ．2.3GFP 报告基因检测法测定病毒滴度同时将0.1、0.01、0.001μL 病毒浓缩液感染2d 后的细胞，置于荧光显微镜下观测．镜检结果显示，GFP 阳性细胞数逐级递减，0.1、0.01、0.001μL 病毒量感染的细胞中，GFP 阳性率分别为52%、6%、0.5%，呈现较好的倍比关系（图3）．通过公式：病毒滴度=感染细胞数×GFP 阳性率×病毒稀释倍数÷病毒量，得出病毒滴度为（8.10±0.79）×109IU/mL ．2.4GFP 报告基因检测法与定量PCR 检测法所得病毒滴度比较，验证定量PCR 检测法准确性通过比较GFP 报告基因检测法和实时定量510152025303510^-110^-210^-3N o r m f l u o r eCycle numberThreshold·····10^410^510^610^710^810^940302010R =0.99980R ^2=0.99960Efficiency=0.96M =0.293B =13.087Concentration （Copy number ）T h r e s h o l d c y c l e396第5期注：病毒感染293T 细胞数为1.5×106.Notes ：293T cells in each well are:1.5×106.图3逐级稀释的慢病毒感染293T 细胞后GFP 的表达情况（A ）0.1μL 病毒感染细胞；（B ）0.01μL 病毒感染细胞；（C ）0.001μL 病毒感染细胞.Fig.3GFP expression in infected 293T cells after 10fold dilution （A ）293T cells infected by 0.1μL virus ；（B ）293T cells infected by 0.01μL virus ；（C ）293T cells infected by 0.001μL virus.PCR 检测法（表2），可以看出，与我们预期结果一致，实时定量PCR 法检测结果与GFP 报告基因检测法检测结果成正相关．多次重复实验结果均表明，两者的检测结果成稳定的正相关性．而且，实时定量PCR 检测法不依赖GFP 蛋白正常表达，对整合入宿主基因组但不能正常表达的病毒仍然能够在其检测范围之内，因此，实时定量PCR 的检测结果更接近实际值，能够更加准确的检测病毒滴度．（A ）（B ）（C）表2GFP 报告基因检测法与定量PCR 检测法测得病毒滴度比较Table 2Comparison of lentivirus titer determined by GFP reporter assay and qRT -PCR approachDose of viral particles6.3×107 6.3×106 6.3×105GFP reporter assay Percentage of GFP +cells Infectious units qRT -PCR assayInfectious units52.0%7.8×1051.3×1066.0%9.0×1042.3×1050.5%7.5×1031.1×1042.5荧光实时定量PCR 法测定病毒实际感染效率通过计算公式：感染效率=有效感染病毒数/感染细胞总病毒颗粒数×100%，得到该实验中慢病毒的感染效率为：2.65±1.07%．3讨论慢病毒载体的研究，目前以HIV -1最为深入．慢病毒载体构建的基本原理是将HIV -1基因组中的基本骨架与编码其功能蛋白相分离，分别改建成载体质粒和表达包装蛋白的质粒，并将两种成分共转染入细胞，从中获得只有一次感染能力而没有复制能力的HIV -1载体假病毒［7］，从而提高了其应用的安全性.近年来，越来越多的研究者利用慢病毒载体系统作为在动物模型研究基因治疗的导入系统，并取得良好的效果［8~10］．与此同时，利用慢病毒载体介导制备转基因动物的研究也得到发展，慢病毒载体介导成功制备了转基因小鼠［11，12］、转基因猪［13］、转基因牛［14］等动物，为基因工程领域的研究奠定基础．由于慢病毒载体的基因转导效率主要取决于病毒滴度，这就使得慢病毒滴度及其感染效率的检测变得很重要．目前，通行的慢病毒滴度检测方法有：1）p24等抗原酶联检测法（p24ELISA 方法）.其缺点是：商业化的ELISA 试剂盒往往太贵，约6000元人民币一盒.而且蛋白含量检测结果不能直接反映其拷贝数；2）使用报告基因系统，无法检测其真实颗粒数及不携带报告基因的假病毒滴度；3）检测逆转录酶活性，用量马海燕等：应用荧光实时定量PCR 方法检测重组慢病毒滴度及其感染效率397生命科学研究2009年大，操作复杂及准确性差．本文所阐述的通过实时定量PCR法测定LTR拷贝数来检测病毒滴度的方法，能在提高慢病毒滴度检测准确性的同时，缩短检测时间、减少检测成本.相比传统检测方法，实时定量PCR 检测法有以下特点：1）由于本检测方法中，实时定量PCR的检测引物设计在慢病毒载体的LTR区，因此检测不依赖于所携带的外源基因；2）传统的GFP报告基因检测方法依赖绿色荧光蛋白的表达，对于外源基因整合入宿主基因组中但由于基因沉默而未能表达GFP蛋白的细胞无法检测，致使滴度测定不能准确地反应病毒的感染效率；而实时定量PCR检测法不依赖GFP报告基因的功能表达，因此其准确程度更高；3）报告基因检测法，因其感染和表达效率随宿主细胞而异，而利用实时定量PCR方法，直接检测病毒的颗粒数和整合入宿主基因组内的外源基因的拷贝数，从而大大提高了其检测的准确性．目前，临床上及实验室所应用的定量PCR方法，通常是检测整合入宿主细胞的基因拷贝数，因此，是对有活力病毒滴度的测定．而本文所介绍的方法，是通过直接裂解病毒，利用荧光实时定量PCR检测总的病毒颗粒数，并结合传统的慢病毒活力滴度的检测方法，对病毒颗粒的感染效率进行检测．所以本方法更适于被应用于研究病毒制备、感染过程中各因素对病毒感染效率的影响．病毒浓缩和感染过程中，由于受超离的压力、反复冻融、受体细胞易感性、病毒自身半衰期等诸多因素影响，使得有效的病毒数要低于其总颗粒数．为了确定实验中病毒的用量，需要预测病毒的实际感染效率．利用本文所介绍的实时定量PCR方法，检测慢病毒颗粒数及有活性的病毒滴度，通过计算有活力的病毒颗粒和总病毒颗粒的比值，我们可以得到每次制备的病毒的感染效率．在我们的实验中，所得到的病毒在293T细胞的实际感染效率为4%左右．而且多次实验表明，反复冻融对病毒感染效率具有很大影响.病毒实际感染效率的测定，为我们在进行具体实验中确定病毒用量具有实际指导意义．经本实验室多次重复试验，结果均表明荧光实时定量PCR法是一种高效、准确、快速的检测重组慢病毒滴度及其感染效率的方法，为重组慢病毒载体的应用奠定了基础．致谢：感谢任兆瑞教授对本文的悉心指导．参考文献（References）:[1]WU X，LI Y，CRISE B，et al.Transcription start regions inthe human genome are favored targets for MLV integration[J].Science，2003，300：1749-1751.[2]DEPALMA M，MONTINI E，SANTONIDESIO F R，et al.Promoter trapping reveals significant differences in integrationsite selection between MLV and HIV vectors in primaryhematopoietic cells[J].Blood，2005，105（6）：2307-2315. [3]SCHRODER A R，SHINN P，CHEN H，et al.HIV-1integration in the human genome favors active genes and localhotspots[J].Cell，2002，110：521-529.[4]JAKOBSSON J，ERICSON C，JANSSON M，et al.Targetedtransgene expression in rat brain using lentiviral vectors[J].Neurosci Res，2003，73（6）：876.[5]张敬之，郭歆冰，谢书阳，等.用慢病毒载体介导产生绿色荧光蛋白（GFP）转基因小鼠[J].自然科学进展（ZHANGJing-zhi，GUO Xin-bin，XIE Shu-yang，et al.Production oftransgenic mice carrying green fluorescence protein gene by alentiviral vector-mediated approach[J].Progress in NatureScience），2006，16（8）：827-832.[6]STEWART S A，OYLOCHOOM D M，PALLISER D，et al．Lentivims-delivered stable gene silencing by RNAi in primarycells[J].RNA，2003，9（4）：493-501．[7]刘茵.慢病毒载体在转基因动物研制中的应用[J].国际遗传学杂志（LIU Yin.Application of lentiviral vectors intransgenic animal development[J].Int J Genet），2007，30（5）：374-377.[8]MAY C，RIVELLA S，CALLEGARI J，et al.Therapeutichaemoglobin synthesis in beta-thalassaemic mice expressinglentivirus-encoded human beta-globin[J].Nature，2000，406（6791）：82-86.[9]HAN X D，LIN C，CHANG J，et al.Fetal gene therapy ofα-thalassemia in a mouse model[J].PNAS，2007，104（21）：9007-9011.[10]LI W，XIE S Y，GUO X B，et al.A novel transgenic mousemodel produced from lentiviral germline integration for thestudy ofβ-thalassemia gene therapy[J].Haematologica，2008，93（3）：357-362.[11]LOIS C，HONG E J，PEASE S，et al.Germline transmissionand tissuespecific expression of transgenes delivered bylentiviral vectors[J].Science，2002，295：868-872.[12]PFEIFER A，IKAWA M，DAYN Y，et al.Transgenesis bylentiviral vectors：lack of gene silencing in mammnlianembryonic stem cells and preimpalntation embryos[J].ProcNatl Acad Sci U S A，2002，99：2140-2145.[13]HOFMANN A，KESSLER B，EWERLING S，et al.Efficienttransgenesis in farm animals by lentiviral emtovora vectors[J].EMBO Reports，2003，4：1054-1060.[14]HOFMANN A，ZAKHARTCHENKO V，WEPPERT M，et al.Generation of transgenic cattle by lentiviral gene transfer intooocytes[J].Biol Reprod，2004，71：405-409.398。

Convergence and Preconditioning Eigenvalue Problems Iterative methods for eigenvalue pro

• Jacobi iteration
A1=D
A2=L+U
– A1-1 is easy to compute (1/ entries along diagonal) – This is easy to compute with the FMM – At element level
• Other classical iterations (Gauss-Seidel, SOR, etc. are harder to write in a way that FMM can be used).
Ax=b → x=A-1b
• Exceptions
– Analytical expression for inverse can be obtained, e.g.,
© Gumerov & Duraiswami, 2003
Direct solutions (2)
• Consumer level – Use LAPACK or MATLAB,
© Gumerov & Duraiswami, 2003
Conjugate Gradient
• Instead of minimizing each time along gradient, choose a set of basis directions to minimize along • Choose these directions to be from the Krylov subspace • Definition: Krylov subspace • Definition: Energy or A-norm of a vector ||x||A=(xt Ax)1/2 • Idea of conjugate gradient method

斯坦福机器视觉CS131

20
12-Mar-16
Special Matrices
• Symmetric matrix
• Skew-symmetric matrix
Fei-Fei Li
Linear Algebra Review
21
12-Mar-16
Outline
• Vectors and matrices
– Basic Matrix Operations – Special Matrices
Fei-Fei Li
Linear Algebra Review
16
12-Mar-16
Matrix Operations
• Transpose – flip matrix, so row 1 becomes column 1
• A useful identity:
Fei-Fei Li
Linear Algebra Review
=
Fei-Fei Li
Linear Algebra Review
9
12-Mar-16
Basic Matrix Operations
• We will discuss:
– Addition – Scaling – Dot product – Multiplication – Transpose – Inverse / pseudoinverse – Determinant / trace
Fei-Fei Li
Linear Algebra Review
19
12-Mar-16
Special Matrices
• Identity matrix I
– Square matrix, 1’s along diagonal, 0’s elsewhere – I ∙ [another matrix] = [that matrix]

哈尔滨工业大学深圳-模式识别-2017-考试重要知识点(word文档物超所值)

λ(αi | ωj ) be the loss incurred for taking action αi when the state of nature is ωj.action αi assign the sample into any class-Conditional risk for i = 1,…,a∑===cj j j j i i x P x R 1)|()|()|(ωωαλαSelect the action αi for which R(αi | x) is minimumR is minimum and R in this case is called the Bayes risk = best reasonable result that can be achieved!λij :loss incurred for deciding ωi when the true state of nature is ωjg i (x) = - R(αi | x)max. discriminant corresponds to min. riskg i (x) = P(ωi | x)max. discrimination corresponds to max. posteriorg i (x) ≡ p(x | ωi ) P(ωi ) g i (x) = ln p(x | ωi ) + ln P(ωi )问题由估计似然概率变为估计正态分布的参数问题极大似然估计和贝叶斯估计结果接近相同，但方法概念不同Please present the basic ideas of the maximum likelihood estimation method and Bayesian estimation method. When do these two methods have similar results ?请描述最大似然估计方法和贝叶斯估计方法的基本概念。

基于PCA方法的BIM构件重建与误差分析

第３６卷第２期
２０２２年４月
Ｖｏｌ．３６Ｎｏ．２
Ａｐｒ．２０２２
粉煤灰综合利用
ＦＬＹＡＳＨＣＯＭＰＲＥＨＥＮＳＩＶＥＵＴＩＬＩＺＡＴＩＯＮ
建筑结构
基于ＰＣＡ方法的ＢＩＭ构件重建与误差分析 ∗
ＢＩＭＣｏｍｐｏｎｅｎｔＲｅｃｏｎｓｔｒｕｃｔｉｏｎａｎｄＥｒｒｏｒＡｎａｌｙｓｉｓｂａｓｅｄｏｎＰＣＡＭｅｔｈｏｄ
和方位参数，用于构件的分类、三维重建和拓扑
分析。
运用ＰＣＡ对点云数据模型进行分析已有许多
研究：王俊、李国俊等［５－６］改进了普通ＰＣＡ算法，
提升了ＰＣＡ方法在处理噪声等粗差方面的能力；
谭凯等［７］基于地面扫描点云的三维模型构建，优
化了辅助点云的分割、滤波与三维表面重建等方
分解后得到的特征值以及特征值之间的关系也不
分主要信息，在数据去噪、数据降维、要素分析
影是一条平面直线，根据ＰＣＡ原理可知，特征值
件分割和提取的既有建筑点云数据，通过对散乱
值 λ ２用于表示法方向点云分布的离散程度，因此
从而描述包括直线、平面、梁和柱等几何图形的
平面，特征值 λ １和 λ ２对应的特征向量ｖ１和ｖ２为平
ｐａｐｅｒｐｒｏｐｏｓｅｓａｃｏｍｐｏｎｅｎｔａｎａｌｙｓｉｓｍｅｔｈｏｄｃｏｍｂｉｎｉｎｇＰｒｉｎｃｉｐａｌＣｏｍｐｏｎｅｎｔＡｎａｌｙｓｉｓ（ＰＣＡ）ａｎｄｓｉｎｇｕｌａｒｖａｌｕｅｄｅｃｏｍｐｏｓｉｔｉｏｎ
（ＳＶＤ）Ｆｉｒｓｔｌｙ，ｔｈｅｄｉｖｅｒｇｅｎｃｅｍａｔｒｉｘｉｓｃａｌｃｕｌａｔｅｄｆｏｒｔｈｅｃｏｍｐｏｎｅｎｔｐｏｉｎｔｃｌｏｕｄｔｈａｔｈａｓｂｅｅｎｒｏｕｇｈｌｙｅｘｔｒａｃｔｅｄ，ａｎｄｔｈｅｎｔｈｅｓｈａｐｅ

Matlab Assignment

Fractal
An endless spiral of little men
x cos θ T = s sin θ y − sin θ x s ⋅ cos θ y + s ⋅ sin θ cos θ
x 1 1 0 x 2 x 1 1 0 x 0 3 T5 = + 1 , T6 = y y + 2 y 3 0 1 3 y 3 0 1 3
Eigenface
任給一張圖，把這些eigenvectors當成basis來重建影像。
projection
reconstruction
Top 10 eigenfaces
Eigenface
輸入參數：
選取輸入影像種類
女性(female) 男性(male) 風景(scenery)
選取輸入影像的編號選取training data的種類
固定pattern旋轉角度與pattern平移大小，改變fractal 旋轉角度。固定fractal旋轉角度與pattern平移大小，改變pattern 旋轉角度。固定pattern旋轉角度與fractal旋轉角度，改變pattern 平移大小。在這三種情況下，各會有怎樣的效果？
透過調整程式的參數，觀察並討論其結果，回答我們所給的問題。
Eigenface
把大小為25x20的face image，想成是一個500維度的vector.
‧‧‧
利用一組face images當成training set，求出其 eigenvectors和eigenvalues. 這裡的eigenvectors就是所謂的eigenfaces.

SCI、EI、ISTP、ISR简介

SCI 、EI 、ISTP 、ISR 简介作者：佚名文章来源：本站原创点击数： 4323 更新时间：2005-7-21SCI 、EI 、ISTP 、ISR 是世界四大重要检索系统，其收录论文的状况是评价国家、单位和科研人员的成绩、水平以及进行奖励的重要依据之一。

我国被四大系统收录的论文数量逐年增长。

学校在"1512工程"建设及科技成果奖励方案中均十分重视四大系统，也已成为教师和科研人员提升自己的努力方向。

《SCI 》（科学引文索引，Science Citation Index ）创刊于1963年，是美国科学情报研究所（ISI ，http://www.isinet.co m ）出版的一部世界著名的期刊文献检索工具。

SCI 收录全世界出版的数、理、化、农、林、医、生命科学、天文、地理、环境、材料、工程技术等自然科学各学科的核心期刊约3500种；扩展版收录期刊5800余种。

ISI 通过它严格的选刊标准和评估程序挑选刊源，而且每年略有增减，从而做到其收录的文献能全面覆盖全世界最重要、最有影响力的研究成果。

所谓最有影响力的研究成果，是指报道这些成果的文献大量地被其它文献引用。

即通过先期的文献被当前文献的引用，来说明文献之间的相关性及先前文献对当前文献的影响力。

这使得SCI 不仅作为一部文献检索工具使用，而且成为对科研进行评价的一种依据。

科研机构被SCI 收录的论文总量，反映出整个学术团体的研究水平、尤其是基础研究的水平；个人的论文被SCI 收录的数量及被引用次数，反映出个人的研究能力与学术水平。

ISI 每年还出版JCR （期刊引用报告，Journal Citation Reports ）。

JCR 对包括SCI 收录的3500种期刊在内的4700种期刊之间的引用和被引用数据进行统计、运算，并针对每种期刊定义了影响因子（Impact Factor ）等指数加以报道。

一种期刊的影响因子，指该刊前二年发表的文献在当年的平均被引用次数。

eigenvector(二个点集之间的匹配)

National Laboratory of Pattern Recognition
Institute of Automation, Chinese Academy of Sciences
模式识别国家重点实验室
中国科学院自动化研究所
National Laboratory of Pattern Recognition
其中 rij 是点 I i 和点 I j 之间的距离.可见H是实对称矩阵,其对角线元素均为1.
National Laboratory of Pattern Recognition
Institute of Automation, Chinese Academy of Sciences
模式识别国家重点实验室
G ij = exp( rij2 / 2σ 2 ),
其中 rij 是点 I i 和点 J j 之间的距离.
National Laboratory of Pattern Recognition
Institute of Automation, Chinese Academy of Sciences
模式识别国家重点实验室
SVD算法的两大准则
相近性准则(Principle of proximity) 排它性准则(Principle of exclusion)
National Laboratory of Pattern Recognition
Institute of Automation, Chinese Academy of Sciences
2000
'Alignment using Spectral Clusters' ---Marco Carcassoni and Edwin R. Hancock, BMVC 2002

物理杂谈 (62)

物理漫談 (S0547) (3,0)物理與哲學、實驗物理-天文學、理論物理-氣體動力學、相對論與重力理論、理論物理-重力理論相對論、凝態物理、高能物理。

Physics and Philosophy, Experimental Physics and Astronomy, Theoretical Physics and thermodynamics, Relativity and Gravity, Gondens-matter Physics, High-energy Physics.光電漫談 (S0640) (3,0)光學與半導體基礎知識。

包含：光電半導體、顯示器裝置、光纖光學及其元件、積體光學、光電積體電路、光儲存裝置、電荷耦合元件及其應用、光子晶體、微光學元件、近場光學、非線性光學、生醫光電等。

Fundamentals of optical and semiconductor; Covers: Photoelectronic semiconductors, Display devices, Fiber optics and its components, Integrated optics, Optoelectronic integrated circuit, Optical storage devices, Charge coupled devices and its application, Photonic crystal, Micro-optical devices, Near field optics, Nonlinear optics, Electro-optics on medicine.天文學 (S0041) (0,3)宇宙概觀、太陽系、星距量測、星的性質、分類與演化、星雲、星團、銀河、系結構分類、宇宙論、天文台及望遠鏡。

Overview of Universe; Solar System; Inter-Stellar Distance; Properties of Stars; Classification and Evolution; Star Nebulae; Star Cluster, Structure and Classification of Glaxies; Cosmology; Observateries and Telescopes.力學(二) (0,3) / 應用力學(二)（0,3）中心力下的運動、多粒子系統動力學、剛體動力學、耦合振動、非線性振動 (選擇)、非慣性參考座標系中的運動 (選擇)、連續系統 (選擇)。

A homotopy method for finding eigenvalues and eigenvectors

i = k first, we get ai ,r = ∑ X j ,i bi , j , r −1 + λi ( 0 ) bi ,i ,r − λi ( 0 ) bi ,i ,r −1 − ∑ ai ,r − mbi ,i ,m . In case
j =1 m =1 ∞ r
i ≠ k , we get
∞
get ai ,rδ ik + ∑ ai ,r − mbi ,k , m = ∑ X j , k bi , j ,r −1 + λk ( 0 ) bi , k ,r − λk ( 0 ) bi ,k , r −1 . The examination of
m =1 j =1
r
∞
this equation splits naturally into the two cases i = k and i ≠ k . Considering the case
j =1 j =1
∞
∞
2
After rearrangement we get λi (θ ) ∑ ϕi , j (θ ) e j = θ ∑ ϕi , j (θ ) Le j + (1 − θ ) ∑ ϕi , j (θ ) Ke j
j =1 j =1 j =1
∞
∞
∞
from which there resultnvectors and eigenvalues of the operator L . DERIVATION We stipulated earlier that {ei }i =1 are eigenvectors of K , whence xi ( 0 ) = ei . We also
λi ( 0 ) = Kei , ei