Projective Normality Of Algebraic Curves And Its Application To Surfaces

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哈尔莫斯测度论pdf

哈尔莫斯测度论pdf哈尔莫斯测度论是数学中的重要理论之一，是测度论的一个分支。

测度论是研究集合的大小的理论。

而哈尔莫斯测度论则是研究无限维向量空间上的测度的理论。

在数学、物理学等领域中，哈尔莫斯测度论有着广泛的应用。

本篇文章将围绕哈尔莫斯测度论的PDF文档来进行讲解，分步骤阐述其重要性和用途。

第一步，介绍哈尔莫斯测度论哈尔莫斯测度论是由丹尼尔·哈尔莫斯创立的测度论分支。

它是在无限维向量空间上研究测度的理论。

哈尔莫斯测度的计算遵循勒贝格积分的规则。

哈尔莫斯测度论的应用包括概率论、函数分析、图像处理、信号处理、量子力学等领域。

哈尔莫斯测度的重要性在于其可以将非负的标量测度扩展到无限维向量空间上，从而使得我们可以在数学和物理学的无限维世界中进行研究。

第二步，论述哈尔莫斯测度论的PDF文档为了方便学者们深入了解哈尔莫斯测度论，许多学者编写了相关的PDF 文档。

这些文档包括哈尔莫斯测度论的定义、性质和计算规则、相关的例题和习题以及研究该领域的历史和前沿进展。

其中，最重要的文献是哈尔莫斯在1966年出版的《测度论》一书。

此外，还有哈尔莫斯测度论的讨论、应用、计算等方面的相关文献。

第三步，分析哈尔莫斯测度论PDF文档的用途在学习和研究哈尔莫斯测度论的过程中，相关的PDF文档对于学者们来说是非常重要的。

首先，这些文献提供了一个全面的概述，以便学者们深入了解哈尔莫斯测度论的定义、性质和应用。

其次，这些文献提供了一些数学和物理学领域中的实际应用，以便学者们了解如何应用哈尔莫斯测度进行量化分析。

最后，这些文献还提供了丰富的例题和习题，以便学者们进行练习和提高。

总之，哈尔莫斯测度论PDF文档对于学习和研究哈尔莫斯测度论来说是非常重要的。

这些文献提供了全面的概述、重要的实际应用以及有用的例题和习题。

希望学者们能够充分利用这些文献，深入了解哈尔莫斯测度论，并在相关领域取得更大的发展和进步。

凯勒度量的里奇形式-概述说明以及解释

凯勒度量的里奇形式-概述说明以及解释1.引言1.1 概述凯勒度量的里奇形式是微分几何中的一个重要概念，它描述了曲率对于度量的影响。

凯勒度量是一种特殊的度量，它满足局部欧几里德性质和全局特殊正则性质。

而里奇形式是通过度量定义的一个与曲率有关的二次微分形式。

在研究从局部到全局几何结构变化的过程中，凯勒度量的里奇形式起到了至关重要的作用。

通过研究里奇形式可以揭示出度量的几何特征，进而推导出曲率的性质。

因此，理解凯勒度量的里奇形式对于研究曲率以及相关的几何问题非常重要。

在本文中，我们将对凯勒度量的定义进行介绍，它是一种满足严格的正则性质的度量。

然后，我们将引入里奇形式的定义，它是通过度量所导出的一个与曲率相关的微分形式。

最后，我们将详细讨论凯勒度量的里奇形式，并探讨它在几何学中的应用。

通过研究凯勒度量的里奇形式，我们可以深入了解曲率对于度量的影响，并从中揭示出各种几何对象的性质和结构。

这对于解决复杂的几何问题以及推动数学和物理学等学科的发展具有重要意义。

在接下来的章节中，我们将进一步阐述凯勒度量的定义和里奇形式的定义，以及它们之间的关系。

通过深入研究这些内容，我们将更加全面地理解凯勒度量的里奇形式以及其在几何学中的重要性。

1.2文章结构文章结构部分的内容可以按照以下方式编写：1.2 文章结构本文将按照以下结构进行阐述和探讨凯勒度量的里奇形式的相关内容：2.正文2.1 凯勒度量的定义首先，我们将介绍凯勒度量的基本概念和定义。

凯勒度量是一种用于度量曲面上各点之间距离的工具，它能够衡量曲面的局部几何特征。

我们将详细解释凯勒度量的数学定义和性质，阐述其在几何学和物理学中的应用。

2.2 里奇形式的定义接下来，我们将引入里奇形式的概念和定义。

里奇形式是一种在黎曼流形上定义的双线性对称张量，它起着衡量黎曼流形曲率的作用。

我们将讨论里奇形式的数学定义、性质和作用，并介绍其在物理学中的应用。

2.3 凯勒度量的里奇形式在这一部分，我们将探讨凯勒度量与里奇形式的关系。

柯西与常微分方程

• 证明思路 • 解的唯一性：假设有两个不同的最大解，那么由局部柯西-利普希茨
定理可以证明其重叠部分的值相同，将两者不同的部分分别延伸在重叠部分上，则会得到一个更“大”的解（只需验证它满足微分方程），矛盾。因此解唯一。
• 解的存在性：证明需要用到佐恩引理，构造所有解的并集。 • 扩展至高阶常微分方程 • 对于一元的高阶常微分方程
Φ(yn)，使得，这样，如果这个序列有一个收敛点 y ，那么y为函数Φ的不动
点，这时就有，于是我们构造出了一个解y。为此，我们从常数函数开始。令这样构造出来的函数列中的每个函数都满足初始条件。并且由于 f 在 U
中满足利普希茨条件，当区间足够小的时候，Φ成为一个收缩映射。根据完备空间的不动点存在定理，存在关于Φ的稳定不动点，于是可知微分方程（1）的解存在。
• 局部唯一性：在包含点t0的足够小的J区间上，微分方程（1）的解是唯一的
（或者说，方程所有的解在足够小的区间上都是重叠的）。
• 这个定理有点像物理学中的决定论思想：当我们知道了一个系统的特性（微
分方程）和在某一时刻系统的情况（x(t0) = x0）时，下一刻的情况是唯一确定的。
• 局部定理的证明 • 一个简洁的证明思路为构造一个总是满足初始条件的函数递归序列yn + 1 =
• 柯西在常微分方程中的主要贡献在于深入
考察并证明了存在唯一性定理。其中主要定理为“柯西－利普希茨定理 ”
此定理最早由柯西于1820年发表，但直到 1868年，才由鲁道夫·利普希茨给出确定的形式。
下面，我们来介绍一下具体的证明过程：
• 局部定理 • 设为一个完备的有限维赋范向量空间（即一个巴拿赫空间），f为一个取值在
–，
• 只需构造向量和相应的映射，就可以使得（2）变为。这时的初始

混合整数二次规划问题的全局最优性条件(英文)

混合整数二次规划问题的全局最优性条件(英文)
李国权;吴至友
【期刊名称】《应用数学》
【年(卷),期】2011(24)4
【摘要】本文给出了混合整数二次规划问题的全局最优性条件,包括全局最优充分性条件和全局最优必要性条件.我们还给出了一个数值实例用以说明如何利用本文所给出的全局最优性条件来判定一个给定点是否是全局最优解.
【总页数】6页(P845-850)
【关键词】全局最优性条件;混合整数二次规划;抽象凸性
【作者】李国权;吴至友
【作者单位】上海大学理学院数学系;巴拉瑞特大学信息技术与数学科学学院【正文语种】中文
【中图分类】O221.2
【相关文献】
1.带LMI约束的混合整数二次规划问题的全局最优性条件 [J], 秦帅;祁云峰;李倩;祁艳妮;
2.带LMI约束的混合整数二次规划问题的全局最优性条件 [J], 秦帅;祁云峰;李倩;祁艳妮
3.一类非凸二次规划问题的全局最优性条件 [J], 张甲;田志远;李敬玉
4.混合整数非线性规划问题的全局最优性条件 [J], 全靖;李国权
5.混合整数二次规划的全局充分性最优条件 [J], 祁云峰;吴至友
因版权原因，仅展示原文概要，查看原文内容请购买。

斯威齐模型概念

斯威齐模型概念解析1. 概念定义斯威齐模型（Pareto distribution），又称为洛伦兹曲线（Lorenz curve），是一种描述不平等分布的概率分布模型。

该模型是由意大利经济学家维尔弗雷多·斯威齐（Vilfredo Pareto）于1896年提出的，用于描述经济和社会领域中收入、财富、权力等指标的分布情况。

斯威齐模型的数学表达为：f(x)=k x k+1其中，x是一个正数，k是一个正参数，f(x)是x的概率密度函数。

2. 关键概念2.1 洛伦兹曲线洛伦兹曲线是斯威齐模型的可视化表示方法，用于展示不平等分布情况。

在洛伦兹曲线上，横坐标表示累积人口或累积收入比例（从小到大排列），纵坐标表示相应累积总收入比例。

通过绘制洛伦兹曲线可以直观地看出不同群体之间收入或财富的分布情况。

2.2 基尼系数基尼系数是衡量不平等程度的指标，通常与洛伦兹曲线一起使用。

基尼系数的取值范围为0到1，数值越大表示不平等程度越高。

基尼系数通过计算洛伦兹曲线下方面积与对角线下方面积的比值得到，公式如下：G=A A+B其中，A表示洛伦兹曲线下方面积，B表示对角线下方面积。

2.3 斯威齐指数斯威齐指数是斯威齐模型中的一个重要参数，用于描述分布的形状。

斯威齐指数越大，说明不平等程度越高。

斯威齐指数可以通过对斯威齐模型进行参数估计得到。

2.4 少部分财富或收入集中现象斯威齐模型描述了财富或收入在社会中的不平等分布情况。

根据该模型，少部分人拥有了大部分的财富或收入，而大多数人只能分享剩余的少部分。

这种现象被称为少部分财富或收入集中现象，也是斯威齐模型的核心概念之一。

3. 重要性3.1 揭示社会不平等问题斯威齐模型能够客观地描述经济和社会领域中财富、收入、权力等指标的分布情况。

通过洛伦兹曲线和基尼系数，可以直观地展示出不同群体之间的不平等程度，揭示出社会中存在的不公平现象。

3.2 政策制定依据斯威齐模型为政策制定提供了重要依据。

Projective Nonnegative Matrix Factorization for Image Compression and Feature Extraction

Projective Nonnegative Matrix Factorization for Image Compression and Feature ExtractionZhijian Yuan and Erkki OjaNeural Networks Research Centre,Helsinki University of Technology,P.O.Box5400,02015HUT,Finland{zhijian.yuan,erkki.oja}@hut.fiAbstract.In image compression and feature extraction,linear expan-sions are standardly used.It was recently pointed out by Lee and Seungthat the positivity or non-negativity of a linear expansion is a very power-ful constraint,that seems to lead to sparse representations for the images.Their technique,called Non-negative Matrix Factorization(NMF),wasshown to be a useful technique in approximating high dimensional datawhere the data are comprised of non-negative components.We proposehere a new variant of the NMF method for learning spatially localized,sparse,part-based subspace representations of visual patterns.The algo-rithm is based on positively constrained projections and is related bothto NMF and to the conventional SVD or PCA decomposition.Two it-erative positive projection algorithms are suggested,one based on mini-mizing Euclidean distance and the other on minimizing the divergence ofthe original data matrix and its non-negative approximation.Experimen-tal results show that P-NMF derives bases which are somewhat bettersuitable for a localized representation than NMF.1IntroductionFor compressing,denoising and feature extraction of digital image windows, one of the classical approaches is Principal Component Analysis(PCA)and its extensions and approximations such as the Discrete Cosine Transform.In PCA or the related Singular Value Decomposition(SVD),the image is projected on the eigenvectors of the image covariance matrix,each of which provides one linear feature.The representation of an image in this basis is distributed in the sense that typically all the features are used at least to some extent in the reconstruction.Another possibility is a sparse representation,in which any given image win-dow is spanned by just a small subset of the available features[1,2,6,10].This kind of representations have some biological signiﬁcance,as the sparse features seem to correspond to the receptiveﬁelds of simple cells in the area V1of the mammalian visual cortex.This approach is related to the technique of Indepen-dent Component Analysis[3]which can be seen as a nongaussian extension of PCA and Factor Analysis.H.Kalviainen et al.(Eds.):SCIA2005,LNCS3540,pp.333–342,2005.c Springer-Verlag Berlin Heidelberg2005334Z.Yuan and E.OjaRecently,it was shown by Lee and Seung[4]that positivity or non-negativity of a linear expansion is a very powerful constraint that also seems to yield sparse representations.Their technique,called Non-negative Matrix Factoriza-tion(NMF),was shown to be a useful technique in approximating high di-mensional data where the data are comprised of non-negative components.The authors proposed the idea of using NMF techniques toﬁnd a set of basis func-tions to represent image data where the basis functions enable the identiﬁcation and classiﬁcation of intrinsic“parts”that make up the object being imaged by multiple observations.NMF has been typically applied to image and text data [4,9],but has also been used to deconstruct music tones[8].NMF imposes the non-negativity constraints in learning the basis images. Both the values of the basis images and the coeﬃcients for reconstruction are all non-negative.The additive property ensures that the components are combined to form a whole in the non-negative way,which has been shown to be the part-based representation of the original data.However,the additive parts learned by NMF are not necessarily localized.In this paper,we start from the ideas of SVD and NMF and propose a novel method which we call Projective Non-negative Matrix Factorization(P-NMF), for learning spatially localized,parts-based representations of visual patterns. First,in Section2,we take a look at a simple way to produce a positive SVD by truncating away negative parts.Section3brieﬂy reviews Lee’s and Seung’s ing this as a baseline,we present our P-NMF method in Section4. Section5gives some experiments and comparisons,and Section6concludes the paper.2Truncated Singular Value DecompositionSuppose that our data1is given in the form of an m×n matrix V.Its n columns are the data items,for example,a set of images that have been vectorized by row-by-row scanning.Then m is the number of pixels in any given image.Typically, n>m.The Singular Value Decomposition(SVD)for matrix V isV=UDˆU T,(1) where U(m×m)andˆU(n×m)are orthogonal matrices consisting of the eigenvectors of VV T and V T V,respectively,and D is a diagonal m×m matrix where the diagonal elements are the ordered singular values of V.Choosing the r largest singular values of matrix V to form a new diagonal r×r matrixˆD,with r<m,we get the compressive SVD matrix X with given rank r,X=UˆDˆU T.(2)1For clarity,we use here the same notation as in the original NMF theory by Lee and SeungProjective Nonnegative Matrix Factorization335 Now both matrices U andˆU have only r columns corresponding to the r largest eigenvalues.The compressive SVD gives the best approximation X of the matrix V with the given compressive rank r.In many real-world cases,for example,for images,spectra etc.,the original data matrix V is non-negative.Then the above compressive SVD matrix X fails to keep the nonnegative property.In order to further approximate it by a non-negative matrix,the following truncated SVD(tSVD)is suggested.We simply truncate away the negative elements byˆX=12(X+abs(X)).(3)However,it turns out that typically the matrixˆX in(3)has higher rank than X. Truncation destroys the linear dependences that are the reason for the low rank. In order to get an equal rank,we have to start from a compressive SVD matrix X with lower rank than the given r.Therefore,toﬁnd the truncated matrix ˆX with the compressive rank r,we search all the compressive SVD matrices X with the rank from1to r and form the corresponding truncated matrices.The one with the largest rank that is less than or equal to the given rank r is the truncated matrixˆX what we choose as theﬁnal non-negative approximation. This matrix can be used as a baseline in comparisons,and also as a starting point in iterative improvements.We call this method truncated SVD(t-SVD).Note that the tSVD only produces the non-negative low-rank approximation ˆX to the data matrix V,but does not give a separable expansion for basis vectors and weights as the usual SVD expansion.3Non-negative Matrix FactorizationGiven the nonnegative m×n matrix V and the constant r,the Nonnegative Matrix Factorization algorithm(NMF)[4]ﬁnds a nonnegative m×r matrix W and another nonnegative r×n matrix H such that they minimize the following optimality problem:minW,H≥0||V−WH||.(4) This can be interpreted as follows:each column of matrix W contains a basis vector while each column of H contains the weights needed to approximate the corresponding column in V using the basis from W.So the product WH can be regarded as a compressed form of the data in V.The rank r is usually chosen so that(n+m)r<nm.In order to estimate the factorization matrices,an objective function deﬁned by the authors as Kullback-Leibler divergence isF=mi=1nµ=1[V iµlog(WH)iµ−(WH)iµ].(5)This objective function can be related to the likelihood of generating the images in V from the basis W and encodings H.An iterative approach to336Z.Yuan and E.Ojareach a local maximum of this objective function is given by the following rules[4,5]:W ia←W iaµV iµ(WH)iµH aµ,W ia←W iajW ja(6)H aµ←H aµi W iaV iµ(WH)iµ.(7)The convergence of the process is ensured2.The initialization is performed using positive random initial conditions for matrices W and H.4The Projective NMF Method4.1Deﬁnition of the ProblemThe compressive SVD is a projection method.It projects the data matrix V onto the subspace of the eigenvectors of the data covariance matrix.Although the truncated method t-SVD outlined above works and keeps nonnegativity, it is not accurate enough for most cases.To improve it,for the given m×n nonnegative matrix V,m<n,let us try toﬁnd a subspace B of R m,and an m×m projection matrix P with given rank r such that P projects the nonnegative matrix V onto the subspace B and keeps the nonnegative property, that is,PV is a nonnegative matrix.Finally,it should minimize the diﬀerence ||V−PV||.This is the basic idea of the Projective NMF method.We can write any symmetrical projection matrix of rank r in the formP=WW T(8) with W an orthogonal(m×r)matrix3.Thus,we can solve the problem by searching for a nonnegative(m×r)matrix W.Based on this,we now introduce a novel method which we call Projective Non-negative Matrix Factorization(P-NMF)as the solution to the following optimality problemminW≥0||V−WW T V||,(9)where||·||is a matrix norm.The most useful norms are the Euclidean dis-tance and the divergence of matrix A from B,deﬁned as follows:The Euclidean distance between two matrices A and B is2The matlab program for the above update rules is available at under the”Computational Neuroscience”discussion category.3This is just notation for a generic basis matrix;the solution will not be the same as the W matrix in NMF.Projective Nonnegative Matrix Factorization337||A−B||2=i,j(A ij−B ij)2,(10) and the divergence of A from BD(A||B)=i,j (A ij logA ijB ij−A ij+B ij).(11)Both are lower bounded by zero,and vanish if and only if A=B.4.2AlgorithmsWeﬁrst consider the Euclidean distance(10).Deﬁne the functionF=12||V−WW T V||2.(12)Then the unconstrained gradient of F for W,∂F∂w ij,is given by∂F∂w ij=−2(VV T W)ij+(WWT VV T W)ij+(VVT WW T W)ij.(13)Using the gradient we can construct the additive update rule for minimization,W ij←W ij−ηij∂F∂w ij(14)whereηij is the positive step size.However,there is nothing to guarantee that the elements W ij would stay non-negative.In order to ensure this,we choose the step size as follows,ηij=W ij(WW T VV T W)ij+(VV T WW T W)ij.(15)Then the additive update rule(14)can be formulated as a multiplicative update rule,W ij←W ij(VV T W)ij(WW T VV T W)ij+(VV T WW T W)ij.(16)Now it is guaranteed that the W ij will stay nonnegative,as everything on the right-hand side is nonnegative.For the divergence measure(11),we follow the same process.First we calcu-late the gradient∂D(V||WW T V)∂w ij=k(W T V)jk+lW lj V ik(17)−kV ik(W T V)jk/(WW T V)ik(18)−k V iklW lj V lk/(WW T V)lk.(19)338Z.Yuan and E.OjaUsing the gradient,the additive update rule becomesW ij ←W ij +ζij ∂D (V ||WW T V )∂w ij(20)where ζij is the step size.Choosing this step size as following,ζij =W ij k V ik [(W T V )jk /(WW T V )ik + l W lj V lk /(WW T V )lk ].(21)we obtain the multiplicative update ruleW ij ←W ij k (W T V )jk + l W lj V ik k V ik ((W T V )jk /(WW T V )ik + l W lj V lk /(WW T V )lk ).(22)It is easy to see that both multiplicative update rules (16)and (22)can ensure that the matrix W is non-negative.4.3The Relationship Between NMF and P-NMFThere is a very obvious relationship between our P-NMF algorithms and the original paring the two optimality problems,P-NMF (9)and the original NMF (4),we see that the weight matrix H in NMF is simply replaced by W T V in our algorithms.Both multiplicative update rules (16)and (22)are obtained similar to Lee and Seung’s algorithms [5].Therefore,the convergence of these two algorithms can also be proved following Lee and Seung [5]by noticing that the coeﬃcient matrix H is replaced by WV .4.4The Relationship Between SVD and P-NMFThere is also a relationship between the P-NMF algorithm and the SVD.For the Euclidean norm,note the similarity of the problem (9)with the conventional PCA for the columns of V .Removing the positivity constraint,this would be-come the usual ﬁnite-sample PCA problem,whose solution is known to be an orthogonal matrix consisting of the eigenvectors of VV T .But this is the matrix U in the SVD of eq.(1).However,now with the positivity constraint in place,the solution will be something quite diﬀerent.5Simulations 5.1Data PreparationAs experimental data,we used face images from the MIT-CBCL database and derived the NMF and P-NMF expansions for them.The training data set con-tains 2429faces.Each face has 19×19=361pixels and has been histogram-equalized and normalized so that all pixel values are between 0and 1.ThusProjective Nonnegative Matrix Factorization339 the data matrix V which now has the faces as columns is361×2429.This matrix was compressed to rank r=49using either t-SVD,NMF,or P-NMF expansions.5.2Learning Basis ComponentsThe basis images of tSVD,NMF,and P-NMF with dimension49are shown in Figure1.For NMF and P-NMF,these are the49columns of the corresponding matrices W.For t-SVD,we show the49basis vectors of the range space of the rank-49nonnegative matrixˆX,obtained by ordinary SVD of this matrix.Thus the basis images for NMF and P-NMF are truly non-negative,while the t-SVD only produces a non-negative overall approximation to the data but does not give a separable expansion for basis vectors and weights.All the images are displayed with the matlab command”imagesc”without any extra scale.Both NMF and P-NMF bases are holistic for the training set. For this problem,the P-NMF algorithm converges about5times faster than NMF.Fig.1.NMF(top,left),t-SVD(bottom,left)and the two versions of the new P-NMF method(right)bases of dimension49.Each basis component consists of19×19pixels340Z.Yuan and E.OjaFig.2.The original face image(left)and its reconstructions by NMF(top row),the two versions of the new P-NMF method under100iterative steps(second and third rows),and t-SVD(bottom row).The dimensions in columns2,3,and4are25,49and 81,respectively5.3Reconstruction AccuracyWe repeated the above computations for ranks r=25,49and81.Figure2 shows the reconstructions for one of the face images in the t-SVD,NMF,and P-NMF subspaces of corresponding dimensions.For comparison,also the original face image is shown.As the dimension increases,more details are recovered. Visually,the P-NMF method is comparable to NMF.The recognition accuracy,deﬁned as the Euclidean distance between the orig-inal data matrix and the recognition matrix,can be used to measure the perfor-mance quantitatively.Figure3shows the recognition accuracy curves of P-NMF and NMF under diﬀerent iterative steps.NMF converges faster,but when the number of steps increases,P-NMF works very similarly to NMF.One thing to be noticed is that the accuracy of P-NMF depends on the initial values.Al-though the number of iteration steps is larger in P-NMF for comparable error with NMF,this is compensated by the fact that the computational complexity for one iteration step is considerably lower for P-NMF,as only one matrix has to be updated instead of two.Projective Nonnegative Matrix Factorization341Fig.3.Recognition accuracies(unit:108)versus iterative steps using t-SVD,NMF and P-NMF with compressive dimension496ConclusionWe proposed a new variant of the well-known Non-negative Matrix Factorization (NMF)method for learning spatially localized,sparse,part-based subspace rep-resentations of visual patterns.The algorithm is based on positively constrained projections and is related both to NMF and to the conventional SVD decompo-sition.Two iterative positive projection algorithms were suggested,one based on minimizing Euclidean distance and the other on minimizing the divergence of the original data matrix and its pared to the NMF method, the iterations are somewhat simpler as only one matrix is updated instead of two as in NMF.The tradeoﬀis that the convergence,counted in iteration steps, is slower than in NMF.One purpose of these approaches is to learn localized features which would be suitable not only for image compression,but also for object recognition. Experimental results show that P-NMF derives bases which are better suitable for a localized representation than NMF.It remains to be seen whether they would be better in pattern recognition,too.342Z.Yuan and E.OjaReferences1. A.Bell and T.Sejnowski.The”independent components”of images are edgeﬁlters.Vision Research,37:3327–3338,1997.2. A.Hyv¨a rinen and P.Hoyer.Emergence of phase and shift invariant features bydecomposition of natural images into independent feature subspaces.Neural Com-putation,13:1527–1558,2001.3. A.Hyv¨a rinen,J.Karhunen,and E.Oja.Independent Component Analysis.Wiley,New York,2001.4. D.D.Lee and H.S.Seung.Learning the parts of objects by non-negative matrixfactorization.Nature,401:788–791,1999.5. D.D.Lee and H.S.Seung.Algorithms for non-negative matrix factorization.InNIPS,pages556–562,2000.6. B.A.Olshausen and D.J.Field.Natural image statistics and eﬃcient coding.Network,7:333–339,1996.7.P.Paatero and U.Tapper.Positive Matrix Factorization:A non-negative factormodel with optimal utilization of error estimations of data values.Environmetrics, 5,111-126,1997.8.T.Kawamoto,K.Hotta,T.Mishima,J.Fujiki,M.Tanaka and T.Kurita.Esti-mation of single tones from chord sounds using non-negative matrix factorization.Neural Network World,3,429-436,July2000.9.L.K.Saul and D.D.Lee.Multiplicative updates for classiﬁcation by mixture mod-ela.In Advances in Neural Information Processing Systems14,2002.10.J.H.van Hateren and A.van der Schaaf.Independent componentﬁlters of natu-ral images compared with simple cells in primary visual cortex.Proc.Royal Soc.London B,265:2315–2320,1998.。

均衡集和凸集

均衡集和凸集
均衡集和凸集是两个不同的概念，但在某些情况下它们可能存在交集。

均衡集，也被称为绝对凸集，是凸集和均衡（或平衡）的集合。

具体来说，如果一个集合既是凸集又是均衡集，那么它被称为均衡凸集。

在数学中，均衡通常指的是对于复数λ（|λ|≤1），无论λ是正值还是负值，都有λA⊂A，即集合A对于一切复数λ保持平衡。

凸集，在凸几何中，是指对于集合内的每一对点，连接该对点的直线段上的每个点也在该集合内。

更具体地说，在欧氏空间中，凸集是对于集合内的每一对点，连接该对点的直线段上的每个点也在该集合内。

例如，立方体是凸集，但是任何中空的或具有凹痕的例如月牙形都不是凸集。

尽管均衡集和凸集的定义和性质不同，但在某些情况下它们可能会重叠。

例如，均衡凸集就是同时满足均衡和凸集两个条件的集合。

这种集合的性质和结构会更加特殊和复杂，需要进一步的研究和探索。

迈克尔·斯皮瓦克微积分

迈克尔·斯皮瓦克微积分
迈克尔·斯皮瓦克（Michael Spivak）是一位知名的数学家和教育家，以他的数学教材和著作而闻名。

他的著作"Calculus"（微积分）是一本广受欢迎的微积分教材，被广泛用于大学和高中的微积分课程。

以下是关于他的微积分著作的一些信息：
1. 书名：《Calculus》（微积分）
2. 作者：迈克尔·斯皮瓦克（Michael Spivak）
3. 出版年份：1967年
4. 内容概述：这本书被认为是一本经典的微积分教材，以其深入的数学严谨性和清晰的解释而著称。

它包括单变量微积分和多变量微积分的内容，并强调数学的证明和推导，帮助学生更深入地理解微积分的概念。

5. 特点：斯皮瓦克的微积分教材以其精炼而深刻的证明和解释闻名，适合对数学有浓厚兴趣和较高数学能力的学生。

它强调数学的严密性和逻辑，鼓励学生进行证明和推理。

6. 使用范围：这本书通常用于大学本科水平的微积分课程，尤其是对于那些数学、工程、物理等专业的学生。

它也可以作为高中或高中课程的高阶微积分材料。

迈克尔·斯皮瓦克的微积分教材在数学教育领域有着广泛的影响，
并被许多教师和学生视为学习微积分的经典教材之一。

这本书以其深度和严谨性而著称，适合那些对深入理解微积分概念有兴趣的学生和教师使用。

有效前沿曲线公式

有效前沿曲线公式
有效前沿曲线（Efficient Frontier）是投资组合理论中的一个重要概念，它描述了在给定的风险水平下，投资者可以期望获得的最大回报。

这条曲线是在均值-方差框架下，通过优化投资组合中各个资产的权重来得到的。

在投资组合理论中，通常假设投资者是理性的，并且希望在给定的风险水平下最大化预期收益，或者在给定的预期收益水平下最小化风险。

有效前沿曲线就是在这样的假设下得到的，它表示了所有可能的投资组合中，风险和预期收益之间的最优权衡关系。

有效前沿曲线的公式通常是通过优化问题来得到的，而不是一个简单的数学表达式。

优化问题的一般形式可以表示为：最大化：预期收益率- 风险厌恶系数* 风险（方差或标准差）
或者
最小化：风险（方差或标准差）
约束条件：预期收益率>= 某个目标值
其中，预期收益率和风险可以用投资组合中各个资产的预期收益率、方差和协方差来计算。

风险厌恶系数是一个反映投资者对风险厌恶程度的参数，它的取值越大，表示投资者对风险的厌恶程度越高。

通过解这个优化问题，可以得到在给定的风险水平下，预期收益最大的投资组合，以及在给定的预期收益水平下，风险最小的投资组合。

这些投资组合构成了有效前沿曲线上的点。

需要注意的是，有效前沿曲线是在一定的假设条件下得到的，实际应用中可能受到多种因素的影响，如交易成本、市场不完全性、投资者偏好等。

因此，在实际应用中，需要根据具体情况进行调整和改进。

棣莫弗中心极限定理

（棣莫弗中心极限定理）De Moivre-Laplace定理，也称为中心极限定理，是概率论的基本结果，该定理指出，在某些情况下，无论单个变量的分布如何，大量独立且分布相同的（即d.）随机变量的总和都将收敛到正态分布。

该定理可以表述如下：设X1、X2、...、Xn是平均μ和标准偏差σ的i.i.d.随机变量序列。

然后，随着n变大，这些随机变量的总和（S）的分布，定义为S = X1 + X2 + ...+ Xn变得越来越正常，平均值为nμ，标准偏差为sqrt(n)σ。

这个定理在许多领域都有重要应用，包括经济学、工程学以及自然和社会科学，在这些领域，它经常被用来模拟对大量兴趣的微小、独立贡献的总和。

例如，如果您试图预测一组学生在测试中的平均分数，您可以使用中心极限定理来模拟分数的分布。

如果个别学生的分数是独立的，分布相同，那么分数之和的分布将大致正常，即使单个分数的分布不是。

这对于进行预测和理解小组分数的可变性非常有用。

AlgebraicGraphTheory

Algebraic Graph TheoryChris Godsil(University of Waterloo),Mike Newman(University of Ottawa)April25–291Overview of the FieldAlgebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example,automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects.One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix.A central topic and important source of tools is the theory of association schemes.An association scheme is,roughly speaking,a collection of graphs on a common vertex set whichﬁt together in a highly regular fashion.These arise regularly in connection with extremal structures:such structures often have an unex-pected degree of regularity and,because of this,often give rise to an association scheme.This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object.Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels.The theory of association schemes provides the most effective means for determining just how large is actually possible;this theory rests on Delsarte’s thesis[4],which showed how to use schemes to translate the problem into a question that be solved by linear programming.2Recent Developments and Open ProblemsBrouwer,Haemers and Cioabˇa have recently shown how information on the spectrum of a graph can be used to proved that certain classes of graphs must contain perfect matchings.Brouwer and others have also investigated the connectivity of strongly-regular and distance-regular graphs.This is an old question,but much remains to be done.Recently Brouwer and Koolen[2]proved that the vertex connectivity of a distance-regular graph is equal to its valency.Haemers and Van Dam have worked on extensively on the question of which graphs are characterized by the spectrum of their adjacency matrix.They consider both general graphs and special classes,such as distance-regular graphs.One very signiﬁcant and unexpected outcome of this work was the construction,by Koolen and Van Dam[10],of a new family of distance-regular graphs with the same parameters as the Grassmann graphs.(The vertices of these graphs are the k-dimensional subspaces of a vector space of dimension v over theﬁniteﬁeld GF(q);two vertices are adjacent if their intersection has dimension k1.The graphs are q-analog of the Johnson graphs,which play a role in design theory.)These graphs showed that the widely held belief that we knew all distance-regular graphs of“large diameter”was false,and they indicate that the classiﬁcation of distance-regular graphs will be more complex(and more interesting?)than we expected.1Association schemes have long been applied to problems in extremal set theory and coding theory.In his(very)recent thesis,Vanhove[14]has demonstrated that they can also provide many interesting results in ﬁnite geometry.Recent work by Schrijver and others[13]showed how schemes could used in combination with semidef-inite programming to provide signiﬁcant improvements to the best known bounds.However these methods are difﬁcult to use,we do not yet have a feel for we might most usefully apply them and their underlying theory is imperfectly understood.Work in Quantum Information theory is leading to a wide range of questions which can be successfully studied using ideas and tools from Algebraic Graph Theory.Methods fromﬁnite geometry provide the most effective means of constructing mutually unbiased bases,which play a role in quantum information theory and in certain cryptographic protocols.One important question is to determine the maximum size of a set of mutually unbiased bases in d-dimensional complex space.If d is a prime power the geometric methods just mentioned provide sets of size d+1,which is the largest possible.But if d is twice an odd integer then in most cases no set larger than three has been found.Whether larger sets exist is an important open problem. 3Presentation HighlightsThe talks mostlyﬁtted into one of four areas,which we discuss separately.3.1SpectraWillem Haemers spoke on universal adjacency matrices with only two distinct eigenvalues.Such matrices are linear combinations of I,J,D and A(where D is the diagonal matrix of vertex degrees and A the usual adjacency matrix).Any matrix usually considered in spectral graph theory has this form,but Willem is considering these matrices in general.His talk focussed on the graphs for which some universal adjacency matrix has only two eigenvalues.With Omidi he has proved that such a graph must either be strong(its Seidel matrix has only two eigenvalues)or it has exactly two different vertex degrees and the subgraph induced by the vertices of a given degree must be regular.Brouwer formulated a conjecture on the minimum size of a subset S of the vertices of a strongly-regular graph X such that no component of X\S was a single vertex.Cioabˇa spoke on his recent work with Jack Koolen on this conjecture.They proved that it is false,and there are four inﬁnite families of counterexamples.3.2PhysicsAs noted above,algebraic graph theory has many applications and potential applications to problems in quantum computing,although the connection has become apparent only very recently.A number of talks were related to this connection.One important problem in quantum computing is whether there is a quantum algorithm for the graph isomorphism problem that would be faster than the classical approaches.Currently the situation is quite open.Martin Roetteler’s talk described recent work[1]on this problem.For our workshop’s viewpoint,one surprising feature is that the work made use of the Bose-Mesner algebra of a related association scheme; this connection had not been made before.Severini discussed quantum applications of what is known as the Lov´a sz theta-function of a graph.This function can be viewed as an eigenvalue bound and is closely related to both the LP bound of Delsarte and the Delsarte-Hoffman bound on the size of an independent set in a regular graph.Severini’s work shows that Lov´a sz’s theta-function provides a bound on the capacity of a certain channel arising in quantum communication theoryWork in quantum information theory has lead to interest in complex Hadamard matrices—these are d×d complex matrices H such that all entries of H have the same absolute value and HH∗=dI.Both Chan and Sz¨o ll˝o si dealt with these in their talks.Aidan Roy spoke on complex spherical designs.Real spherical designs were much studied by Seidel and his coworkers,because of their many applications in combinatorics and other areas.The complex case languished because there were no apparent applications,but now we have learnt that these manifest them-selves in quantum information theory under acronyms such as MUBs and SIC-POVMs.Roy’s talk focussedon a recent 45page paper with Suda [12],where (among other things)they showed that extremal complex designs gave rise to association schemes.One feature of this work is that the matrices in their schemes are not symmetric,which is surprising because we have very few interesting examples of non-symmetric schemes that do not arise as conjugacy class schemes of ﬁnite groups.3.3Extremal Set TheoryCoherent conﬁgurations are a non-commutative extension of association schemes.They have played a sig-niﬁcant role in work on the graph isomorphism problem but,in comparison with association schemes,they have provided much less information about interesting extremal structures.The work presented by Hobart and Williford may improve matters,since they have been able to extend and use some of the standard bounds from the theory of schemes.Delsarte [4]showed how association schemes could be used to derive linear programs,whose values provided strong upper bounds on the size of codes.Association schemes have both a combinatorial structure and an algebraic structure and these two structures are in some sense dual to one another.In Delsarte’s work,both the combinatorial and the algebraic structure had a natural linear ordering (the schemes are both metric and cometric)and this played an important role in his work.Martin explained how this linearity constraint could be relaxed.This work is important since it could lead to new bounds,and also provide a better understanding of duality.One of Rick Wilson’s many important contributions to combinatorics was his use of association schemes to prove a sharp form of the Erd˝o s-Ko-Rado theorem [15].The Erd˝o s-Ko-Rado theorem itself ([5])can certainly be called a seminal result,and by now there are many analogs and extensions of it which have been derived by a range of methods.More recently it has been realized that most of these extensions can be derived in a very natural way using the theory of association schemes.Karen Meagher presented recent joint work (with Godsil,and with Spiga,[8,11])on the case where the subsets in the Erd˝o s-Ko-Rado theorem are replaced by permutations.It has long been known that there is an interesting association scheme on permutations,but this scheme is much less manageable than the schemes used by Delsarte and,prior to the work presented by Meagher,no useful combinatorial information had been obtained from it.Chowdhury presented her recent work on a conjecture of Frankl and F¨u redi.This concerns families F of m -subsets of a set X such that any two distinct elements of have exactly λelements in common.Frankl and F¨u redi conjectured that the m -sets in any such family contain at least m 2 pairs of elements of X .Chowdhury veriﬁed this conjecture in a number of cases;she used classical combinatorial techniques and it remains to see whether algebraic methods can yield any leverage in problems of this type.3.4Finite GeometryEric Moorhouse spoke on questions concerning automorphism groups of projective planes,focussing on connections between the ﬁnite and inﬁnite case.Thus for a group acting on a ﬁnite plane,the number of orbits on points must be equal to the number of orbits on lines.It is not known if this must be true for planes of inﬁnite order.Is there an inﬁnite plane such that for each positive integer k ,the automorphism group has only ﬁnitely many orbits on k -tuples?This question is open even for k =4.Simeon Ball considered the structure of subsets S of a k -dimensional vector space over a ﬁeld of order q such that each d -subset of S is a basis.The canonical examples arise by adding a point at inﬁnity to the point set of a rational normal curve.These sets arise in coding theory as maximum distance separable codes and in matroid theory,in the study of the representability of uniform matroids (to mention just two applications).It is conjectured that,if k ≤q −1then |S |≤q +1unless q is even and k =3or k =q −1,in which case |S |≤q +2.Simeon presented a proof of this theorem when q is a prime and commented on the general case.He developed a connection to Segre’s classical characterization of conics in planes of odd order,as sets of q +1points such that no three are collinear.There are many analogs between ﬁnite geometry and extremal set theory;questions about the geometry of subspaces can often be viewed as q -analogs of questions in extremal set theory.So the EKR-problem,which concerns characterizations of intersecting families of k -subsets of a ﬁxed set,leads naturally to a study of intersecting families of k -subspaces of a ﬁnite vector space.In terms of association schemes this means we move from the Johnson scheme to the Grassmann scheme.This is fairly well understood,with thebasic results obtained by Frankl and Wilson[6].But inﬁnite geometry,polar spaces form an important topic. Roughly speaking the object here is to study the families of subspaces that are isotropic relative to some form, for example the subspaces that lie on a smooth quadric.In group theoretic terms we are now dealing with symplectic,orthogonal and unitary groups.There are related association schemes on the isotropic subspaces of maximum dimension.Vanhove spoke on important work from his Ph.D.thesis,where he investigated the appropriate versions of the EKR problem in these schemes.4Outcome of the MeetingIt is too early to offer much in the way of concrete evidence of impact.Matt DeV os observed that a conjecture of Brouwer on the vertex connectivity of graphs in an association scheme was wrong,in a quite simple way. This indicates that the question is more complex than expected,and quite possibly more interesting.That this observation was made testiﬁes to the scope of the meeting.On a broader level,one of the successes of the meeting was the wide variety of seemingly disparate topics that were able to come together;the ideas of algebraic graph theory touch a number of things that would at ﬁrst glance seem neither algebraic nor graph theoretical.There was a lively interaction between researchers from different domains.The proportion of post-docs and graduate students was relatively high.This had a positive impact on the level of excitement and interaction at the meeting.The combination of expert and beginning researchers created a lively atmosphere for mathematical discussion.References[1]A.Ambainis,L.Magnin,M.Roetteler,J.Roland.Symmetry-assisted adversaries for quantum state gen-eration,arXiv1012.2112,35pp.[2]A.E.Brouwer,J.H.Koolen.The vertex connectivity of a distance-regular graph.European bina-torics30(2009),668–673.[3]A.E.Brouwer,D.M.Mesner.The connectivity of strongly regular graphs.European binatorics,6(1985),215–216.[4]P.Delsarte.An algebraic approach to the association schemes of coding theory.Philips Res.Rep.Suppl.,(10):vi+97,1973.[5]P.Erd˝o s,C.Ko,R.Rado.Intersection theorems for systems ofﬁnite sets.Quart.J.Math.Oxford Ser.(2),12(1961),313–320.[6]P.Frankl,R.M.Wilson.The Erd˝o s-Ko-Rado theorem for vector binatorial Theory,SeriesA,43(1986),228–236.[7]D.Gijswijt,A.Schrijver,H.Tanaka.New upper bounds for nonbinary codes based on the Terwilligeralgebra and semideﬁnite binatorial Theory,Series A,113(2006),1719–1731. [8]C.D.Godsil,K.Meagher.A new proof of the Erd˝o s-Ko-Rado theorem for intersecting families of per-mutations.arXiv0710.2109,18pp.[9]C.D.Godsil,G.F.Royle.Algebraic Graph Theory,Springer-Verlag,(New York),2001.[10]J.H.Koolen,E.R.van Dam.A new family of distance-regular graphs with unbounded diameter.Inven-tiones Mathematicae,162(2005),189-193.[11]K.Meagher,P.Spiga.An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q)acting onthe projective line.arXiv0910.3193,17pp.[12]A.P.Roy,plex spherical Codes and designs,(2011),arXiv1104.4692,45pp.[13]A.Schrijver.New code upper bounds from the Terwilliger algebra and semideﬁnite programming.IEEETransactions on Information Theory51(2005),2859–2866.[14]F.Vanhove.Incidence geometry from an algebraic graph theory point of view.Ph.D.Thesis,Gent2011.[15]R.M.Wilson.The exact bound in the Erds-Ko-Rado binatorica,4(1984),247–257.。

乘积空间的里奇曲率

乘积空间的里奇曲率1.引言1.1 概述在微分几何领域中，研究乘积空间的里奇曲率是一项重要而又有意义的工作。

乘积空间可以被简单地定义为两个或多个空间的直积，它的性质和结构在许多领域中都有广泛的应用。

里奇曲率是描述空间弯曲程度的一个重要指标。

里奇曲率的计算方法是通过测量空间中的切向量场之间的变化率来得到的。

它提供了一种量化空间曲率的方法，帮助我们理解空间的性质和结构。

本文的目的是深入研究乘积空间的里奇曲率，并探讨其对空间性质的影响。

通过对乘积空间的定义和性质进行介绍，我们将探讨如何计算乘积空间的里奇曲率，并分析其在实际问题中的应用和意义。

在第二节中，我们将详细讨论乘积空间的定义和性质。

乘积空间的定义是基于直积的概念，我们将解释直积是怎样构建出乘积空间的，并介绍乘积空间的性质，例如维度、拓扑结构等。

第三节将着重讨论里奇曲率的概念和计算方法。

我们将介绍里奇曲率是如何通过测量空间中的切向量场之间的变化率来得到的，并探讨计算里奇曲率的常用方法和技巧。

最后，我们将在结论部分总结本文的研究成果，并探讨乘积空间的里奇曲率对空间性质的影响。

乘积空间的里奇曲率可以帮助我们理解空间的曲率分布情况，从而对空间结构和性质进行分析和应用。

通过本文的研究，我们希望能够增进对乘积空间的理解，并为相关领域的研究工作提供一定的指导和参考。

同时，我们也期望可以进一步探索乘积空间里奇曲率的应用潜力，为更广泛的领域提供理论和实践的支持。

1.2 文章结构文章结构的主要目的是为读者提供一种清晰的框架，以便他们可以更好地理解和组织阅读过程。

本文的结构包括以下几个部分：1. 引言：本部分将对乘积空间的里奇曲率进行简要概述，并介绍文章的结构和目的。

通过引言，读者可以了解到本文讨论的主要内容和目标。

2. 正文：本部分将详细介绍乘积空间的定义和性质，以及里奇曲率的概念和计算方法。

通过对乘积空间和里奇曲率的深入探讨，读者可以建立对相关概念和理论的基本认识。

用数学来说我爱你英语作文

Mathematics,a language of universality and precision,can be used to express the most profound emotions,such as love.When we say I love you in English,we are conveying a message that is deeply personal and heartfelt.However,when we translate this into the language of mathematics,it takes on a different kind of beauty and depth. Heres how you might express I love you using mathematical concepts and symbols:1.The Infinity Symbol:The infinity symbol is a powerful representation of love because it suggests that the feeling is boundless and eternal.In a mathematical love letter,you could write,My love for you is like,it has no end.2.Piπ:Pi is an irrational number,which means it goes on forever without repeating.This could be a metaphor for the endless love one has for another.You might say,My love for you is likeπ,it continues infinitely without repeating.3.The Golden Ratioφ:The golden ratio,approximately1.618,is found in nature and is associated with beauty and balance.It could be used to express the harmony in a relationship.Our love is like the golden ratio,perfectly balanced and beautiful.4.The Pythagorean Theorem a²b²c²:This theorem is a fundamental principle in geometry and could be used to symbolize the foundation of a relationship.Just as a²b²c²,our love is the sum of our individual strengths,creating a strong bond.5.The Fibonacci Sequence:This sequence starts with0and1,and each subsequent number is the sum of the two preceding ones.It could represent the growth of love over time.Our love is like the Fibonacci sequence,growing and evolving with each passing day.6.Eulers Identity eiπ10:This equation is considered one of the most beautiful in mathematics.It combines five fundamental constants e,i,π,1,and0in a simple equation. Our love is an equation as beautiful as Eulers identity,a perfect balance of the complex and the simple.7.The Heart Curve r a1sinθ:This polar equation describes a shape that looks like a heart.Its a direct mathematical representation of love.Our love is like the heart curve,a perfect shape that is uniquely ours.8.The Lorenz Attractor:This is a set of chaotic solutions to the Lorenz system of differential equations,which can be visualized as a butterfly shape.It could symbolize the unpredictable yet beautiful nature of love.Our love is like the Lorenz Attractor, chaotic yet beautiful,always returning to the same beautiful pattern.In conclusion,while I love you in English is a straightforward declaration,using mathematics to express this sentiment adds a layer of complexity and depth that can make the message even more special and meaningful.Its a way to show that love is not just an emotion but also a fundamental part of the universe,as intricate and as infinite as the mathematics that describe it.。

the 可积哈密顿系统知识天地guide download

週報第1228期1知識天地可積哈密頓系統謝仲副研究員（數學研究所）實際上在日常生活或自然現象中，我們常用方程式來描述現象，為了瞭解現象我們試著將方程式的解求出來，並將解盡可能以簡單的方式表現出來。

當我們在運動場上，看到一顆足球在滾動，你可能會說這顆足球直接的滾向球門。

當你想對這顆球的運動做出較精確的描述，也許你會記錄出這顆球隨時間的消逝而做出的位置、速度、加速度……的變化，將這些變化的相互牽連寫出來，就成了描述這顆球的運動系統方程式。

因此對於一個系統是否可解出來，也是瞭解一個系統的重要指標。

從傳統的觀點，我們大都是透過積方的方式（integration by quadrature ）得到系統的解，因此很多時候，我們說一個系統是可積的系統（integrable system ），也就意味著，這個系統可確解（exactly solvable system ）。

從古早以來數學家對於尋找可積系統一直非常感興趣，藉助可積系統的研究，用來描述古典系統和量子系統是理論物理的數學核心。

去尋找一個一致性的可積性的定義是非常困難的。

尤拉（Euler ）和拉格朗日（Lagrange ）建立了牛頓力學的數學架構，雅可比（Jacobi ）將費瑪（Fermat ）最少時間原則（Fermat’s principle of least time ）和哈密頓（Hamilton ）最小運動（least action ）的觀念運用在動態系統，得到了哈密頓動態系統（Hamiltonian System ）表示法。

在這個架構下，我們可以探討劉維爾（Liouville ）有關可積性的想法。

我們將不從數學或物理學的角度去嚴謹的切入，而從一般常識性或直覺性用類比的方式去看，在這個想法後面我們想表達的是什麼。

一般來說在辛流形上的一個光滑函數H 可以用來定義一個哈密頓系統，通常這個函數稱為哈密頓函數或能量函數。

我們可想像這個函數描述了在相空間上的能量或訊息的分佈。

棣莫弗-拉普拉斯中心极限定理的英文

棣莫弗-拉普拉斯中心极限定理的英文全文共四篇示例，供读者参考第一篇示例：The Central Limit Theorem (CLT) is a fundamental concept in the field of statistics that was first formulated by Pierre-Simon Laplace in the late 18th century. The theorem states that the distribution of the sum (or average) of a large number of independent and identically distributed random variables approaches a normal distribution regardless of the shape of the original distribution.第二篇示例：The Central Limit Theorem (CLT) is a fundamental concept in statistics and probability theory. It states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the original distribution of the variables. One of the most famous versions of this theorem is the De Moivre-Laplace Central Limit Theorem, named after two prominent mathematicians, Abraham de Moivre andPierre-Simon Laplace.第三篇示例：The central limit theorem (CLT) is a fundamental theorem in probability theory and statistics that states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the shape of the original distribution. This theorem has profound implications in various fields such as finance, engineering, and biology, where it allows researchers to make reliable predictions even when they have limited information about the underlying data.第四篇示例：The Central Limit Theorem, also known as the DeMoivre-Laplace Theorem, is a fundamental concept in probability theory and statistics. It states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the distribution of the original variables.。

克拉夫特-麦克米伦定理

克拉夫特-麦克米伦定理
克拉夫特-麦克米伦定理（Cayley–McMullen theorem）是一个在组合数学中应用的重要定理，它描述了一个特定类型的多面体的面数。

该定理由亚瑟·凯利（Arthur Cayley）和彼得·麦克米伦（Peter McMullen）在1965年提出。

定理的陈述如下：对于一个凸多面体，如果它的每个面都是一个正多边形，并且每个多边形的边数都是一个素数，那么这个多面体最多只能有4个面。

这个定理的证明基于对凸多面体的面数、边数和顶点数之间的关系的研究。

具体而言，凸多面体的面数、边数和顶点数之间满足欧拉公式：F + V = E + 2，其中F表示面数，V表示顶点数，E表示边数。

根据欧拉公式，结合题设条件，可以得到一个关于面数的方程，进而推导出最多只能有4个面的结论。

克拉夫特-麦克米伦定理在组合数学研究中有广泛应用，特别是在多面体和多面立体的研究中。

它为研究六面体、四面体等具有特殊性质的多面体提供了重要的理论依据。

贝尔曼最优公式的证明

贝尔曼最优公式的证明贝尔曼最优公式，又称为贝尔曼方程或贝尔曼方程组，是马尔可夫决策过程中的重要概念。

它是由美国数学家Richard E. Bellman在20世纪中期提出的，被广泛应用于优化问题和强化学习中。

贝尔曼最优公式的证明是一个精妙而复杂的过程，本文将以人类的视角进行叙述，让读者更好地理解和感受这个公式的含义。

我们来介绍一下马尔可夫决策过程。

马尔可夫决策过程是一种数学模型，用于描述具有随机性的决策问题。

在这个模型中，决策问题可以被分解为一系列的状态和决策，每个状态都有一个关联的价值，决策的目标就是使得累计的价值最大化。

贝尔曼最优公式就是用来计算每个状态的最优价值的。

假设我们有一个马尔可夫决策过程，其中包含有限个状态和决策。

我们定义一个函数V(s)，表示在状态s下的最优价值。

贝尔曼最优公式的核心思想就是，一个状态的最优价值等于该状态下所有可能决策的预期价值的最大值。

具体来说，对于每个状态s，我们考虑在该状态下的所有可能决策。

假设我们选择了某个决策a，那么在下一个状态s'的最优价值就是V(s')。

根据马尔可夫性质，s'的最优价值又等于V(s')。

所以在状态s 下，决策a的预期价值就是当前的收益加上下一个状态的最优价值，即R(s,a) + V(s')。

由于我们不知道下一个状态是什么，只能根据概率分布来估计。

所以我们将所有可能的下一个状态的最优价值乘以对应的概率，并将它们相加，得到决策a在状态s下的预期价值。

这个过程可以用一个求和符号来表示。

状态s的最优价值V(s)就是在所有可能决策的预期价值中取最大值。

这个过程可以用一个求最大值的运算来表示。

通过不断迭代计算，我们可以逐步求得所有状态的最优价值。

最后，我们就可以根据最优价值来选择最优决策，使得累计的价值最大化。

贝尔曼最优公式的证明过程虽然复杂，但它的思想是清晰而直观的。

它告诉我们，在一个马尔可夫决策过程中，我们可以通过计算每个状态的最优价值来得到最优决策。

闵可夫斯基几何

闵可夫斯基几何
维纳·闵可夫斯基几何是一种在20世纪初兴起的早期的几何理论，它被用于描述有限的空间，研究在它自身拓扑学上的性质并且允许从拓扑学出发来推断和解释空间，这种几何，以其特有的映射方式定义全局性特征，包括基数，空间正则性和拓扑结构，是一种重要的几何形式。

维纳·闵可夫斯基几何的早期应用围绕着它的拓扑学方面展开，这是由它的独特结构，特别是由它的拓扑结构性质所产生的，同时它又能深刻表达所有拓扑结构与空间特性，例如，在这种几何上概括基数及其向量场。

在互联网时代，维纳·闵可夫斯基几何占据了更重要的地位。

这一几何理论可以非常有效地概括出客观世界中所有系统、网络的特性，同时又能运用其结构信息构建基本模型，并被广泛应用于信息的网络归类，特殊点的检测、信息的传播及演化，以及复杂系统的分析和表达。

它有助于人们对于互联网和它背后的结构及性质有更深的见解，并有助于增强对新信息的な把握。

除此之外，维纳·闵可夫斯基几何还可用于处理具有复杂结构的复杂数据，例如，它可以有效运用于社交网络分析，帮助我们优化聚类及其他复杂信息处理，以及其他复杂信息系统的分析，从而避免使用复杂的线性代数工具或先进数据分析工具。

维纳·闵可夫斯基几何为我们提供了一种非常方便的框架，基于这种框架，我们可以方便地探究网络的特征、特性和结构，也可以深入地解读复杂的数据，以及使用维纳·闵可夫斯基几何驱动的技术来进行高效算法开发、计算机建模和数据分析。

维纳·闵可夫斯基几何是对当今社会信息处理未来的一个期许。

霍普夫-庞加来指标定理 -回复

霍普夫-庞加来指标定理-回复霍普夫庞加来指标定理（Hopf-Poincaréindex theorem）是拓扑数学中的一个重要定理，它与微分流形上的向量场的振荡性质相关。

霍普夫夫庞加来指标定理在流形上的奇异点的研究中起到了重要的作用，它被广泛运用于动力系统和控制论等领域。

本文将逐步介绍这个定理，并解释其在数学和其他学科中的重要性。

第一步：引言我们首先来简单介绍一下霍普夫庞加来指标定理。

该定理是由德国数学家海因利希·霍普夫（Heinz Hopf）和法国数学家亨利·庞加来（Henri Poincar é）在20世纪初分别独立提出的。

它提供了一个关于微分流形上向量场奇异点指标的非平凡性质的判断方法。

第二步：微分流形与向量场为了理解这个定理，我们需要先了解一些基本概念。

微分流形是一个可以与欧几里德空间中的点进行一一对应的集合，它具有平滑曲线的性质。

在微分流形上定义了向量场，即在每一点上都存在一个切向量的分布。

向量场可以用来描述流体的流动、力的分布等现象。

第三步：奇异点和振荡性质在微分流形上的向量场中，奇异点是指向量场变为零向量的点。

在奇异点附近，向量场的振荡性质成为研究的重点。

霍普夫庞加来指标定理提供了一种方法来确定奇异点的振荡性质。

第四步：霍普夫指标霍普夫指标是一种用来描述向量场奇异点振荡性质的数值。

对于一个二维流形上的向量场，奇异点可以被分为两类：吸引奇异点和驱离奇异点。

吸引奇异点是指向量场在该点附近的流线在时间推移中趋近于该点的奇异点，而驱离奇异点则相反。

霍普夫指标可以用来判断奇异点是吸引奇异点还是驱离奇异点，从而确定振荡性质。

第五步：霍普夫庞加来指标定理的表述霍普夫庞加来指标定理可以被描述为：对于一个具有有限个奇异点的二维微分流形上的向量场，可以通过计算各个奇异点的霍普夫指标之和来判断整个向量场的振荡性质。

如果该指标之和为正，则向量场存在奇数个驱离奇异点和零个或偶数个吸引奇异点，如果指标之和为负，则存在偶数个驱离奇异点和零个或奇数个吸引奇异点。

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a r X i v :m a t h /0601189v 1 [m a t h .A G ] 9 J a n 2006PROJECTIVE NORMALITY OF ALGEBRAIC CUR VESAND ITS APPLICATION TO SURF ACESSEONJA KIM AND YOUNG ROCK KIMAbstract.Let L be a very ample line bundle on a smooth curve C of genus g with 3g +36−2h 1(C,L )}.Let C be a triple covering of genus p curve C ′with C φ→C ′and D a divisor on C ′with4p <deg D <g −12<deg L ≤2g −5is normallygenerated for deg L >max {2g +2−4h 1(C,L ),2g −g −16−2p .It is a kind of generalization of the result2SEONJA KIM AND YOUNG ROCK KIMthat K C(−rg13)on a trigonal curve C is normally generated for3r≤g2,6e−∆}<f−1<∆−2e−62∆−e if its general hyperplane section H is linearly normal,where∆:=deg S−r+1.Furthermore we characterize smooth projective surfaces S for K2S=−2f−e+2,0(cf.Theorem3.2).These applications were derived by the methods in Akahori’s,Livorni’s and Sommese’s papers ([2],[9],[12]).We follow most notations in[1],[4],[6].Let C be a smooth irreducible projective curve of genus g≥2.The Cliﬀord index of C is taken to be Cliﬀ(C)=min{Cliﬀ(L)|h0(C,L)≥2,h1(C,L)≥2},where Cliﬀ(L)= deg L−2(h0(C,L)−1)for a line bundle L on C.By abuse of notation,we sometimes use a divisor D on a smooth variety V instead of O V(D).We also denote H i(V,O V(D))by H i(V,D)and h0(V,L)−1by r(L)for a line bundle L on V.We denote K V a canonical line bundle on a smooth variety V.2.Normal generation of a line bundle on a smooth curve Any line bundle of degree at least2g+1on a smooth curve of genus g is normally generated.If the degree is at most2g,then there are curves which have a non normally generated line bundle of given degree([8],[10],[11]). In this section,we investigate the normal generation of a line bundle with given degree on a smooth curve under some condition about the speciality of the line bundle.Theorem2.1.Let L be a very ample line bundle on a smooth curve C of genus g with3g+36−2h1(C,L)}.Proof.Suppose L is not normally generated.Then there exists a line bundle A≃L(−R),R>0,such that(i)Cliﬀ(A)≤Cliﬀ(L),(ii)deg A≥g−1PROJECTIVE NORMALITY OF ALGEBRAIC CURVES AND ITS APPLICATION 3C?π:projectionφN 2φN 1C 1where C i =φN i (C ).If we set m i :=deg φN i ,i =1,2,then we have m 2|m 1.If N 1is bira-tionally very ample,then by Lemma 9in [8]and deg K C L −1<g −15.It is a contradiction to deg L >2g +2−4h 1(C,L )that is equivalent to deg K C L −1<4(h 0(C,K C L −1)−1).Therefore N 1isnot birationally very ample,and then we have m 1≤3since deg K C L −1<4(h 0(C,K C L −1)−1).Set H 1be a hyperplane section of C 1.If |H 1|on a smooth model of C 1is special,then r (N 1)≤deg N 12,since N 2(−G )∼=N 1andCliﬀ(N 2)≤Cliﬀ(A )≤Cliﬀ(L )=Cliﬀ(N 1).In case deg N 2≥g we have n ≤2deg N 2−g +13=2g −2−deg N 26,since N 2=K C A −1(−B 2)and deg A ≥g −16−2h 1(C,L )is equivalent to Cliﬀ(K C L −1)<g −14SEONJA KIM AND YOUNG ROCK KIMnumberπ(d,r)has the propertyπ(d,r)≤π(d−2,r−1)for d≥3r−2and r≥3,whereπ(d,r)=m(m−1)2)≤π(deg N1,r(N1)),because of2≤r(N1)≤r(N2)−deg G4and deg N1<g−16.Thus the condition deg K C L−1<4(h0(C,K C L−1)−1)yields the following inequalities:2deg N22,which contradicts to N1 N2.Accordingly|H2|is also nonspecial. Now we have r(N i)=deg N i3+2p=Cliﬀ(N1)≥Cliﬀ(N2)=deg N26−2p.Then K C(−φ∗D)becomes a very ample line bundle which is normally generated.Proof.Set d:=deg D and L:=K C(−φ∗D).Suppose L is not base point free,then there is a P∈C such that|K C L−1(P)|=g r+13d+1.Note thatg r+1 3d+1cannot be composed withφby degree reason.Therefore we haveg≤6d+3p due to the Castelnuovo-Severi inequality.Hence it cannot occur by the condition d<g−16−2p produces Cliﬀ(K C L−1)=d+2p<g−16−2h1(C,L)is satisﬁed.The condition4p<d induces deg K C L−1>4(h0(C,K C L−1)−1),i.e.,deg L>2g+2−4h1(C,L).Consequently L is normally generated by Theorem2.1.PROJECTIVE NORMALITY OF ALGEBRAIC CURVES AND ITS APPLICATION5 Remark2.3.In fact,we have similar result in[8]for trigonal curve C: KC(−rg13)is normally generated if3r<g2,6e−∆}<f−1<∆−2e−62∆−e. Proof.From the linear normality of H,we get h0(H,O H(1))=r and hence h1(H,O H(1))=−deg O H(1)−1+g(H)+h0(H,O H(1))=−2∆+e−1+g(H)+h0(H,O H(1))=g(H)−∆=fTherefore we have h1(H,O H(1))>deg K H⊗O H(−1)2+1anddeg O H(1)=2∆−e=2g(H)−2−(2f+e−2).Thus O H(1)satisﬁes deg O H(1)>2g(H)+2−4h1(H,O H(1)).The condition f−1>6e−∆implies deg O H(1)>2g−g−13yields deg O H(1)>3g+32.If we consider the adjunction formula g(H)=K S.H+H.H2∆−e by theHodge index theorem K2S H2≤(K S.H)2.Hence the theorem is proved. Assume that(2f+e−2)2<2∆−e in the above theorem,then we have −2f−e+2≤K2S≤0.Observe the cases for K2S=−2f−e+2or0,then we obtain the following result by using similar method in[2].6SEONJA KIM AND YOUNG ROCK KIMProposition3.2.Let S satisfy the conditions in Theorem3.1.Then S is a minimal elliptic surface of Kodaira dimension1if K2S=0and|K S|has noﬁxed component.Also S is a surface blown up at2f+e−2points on a K3surface in case K2S=−2f−e+2.Proof.Assume|K S|has noﬁxed component with K2S=0.Then S is minimal by adjunction formula and useful remark III.5in[3].Also the Kodaira dimensionκof S is at most one since K2S≤0.Since p g>1,S is nonruled and soκ≥0.Ifκ=0then p g≤1by Theorem VIII.2in[3]and thusκmust be1.Hence by Proposition IX.2in[3]there is a smooth curve B and a surjective morphism p:S→B whose genericﬁbre is an elliptic curve which means that S is a minimal elliptic surface of Kodaira dimension 1.If K2S=−2f−e+2.LetφH+KS=s◦r be the Remmert-Stein factorizationofφH+KS andˆS=r(S).Then we can use Propositon2.0.6in[9]as statedin the proof of the previous theorem.And we obtainH2−K2S=(2∆−e)−(−2f−e+2)=2g(H)−2,which yieldsˆS is a minimal model and KˆS =0,in other words,ˆS is a K3surface by using Propositon2.0.6(iv-1)in[9].Also by Propositon2.0.6(ii) in[9],S is a surface blown up at2f+e−2points on a K3surfaceˆS since ˆd−d=2g(H)−2−(2∆−e)=2f+e−2.References[1]Arbarello,E.,Cornalba,M.,Griﬃths P.A.and Harris J.,Geometry of AlgebraicCurves I,Springer Verlag,1985.[2]Akahori,K.,Classiﬁcation of projective surfaces and projective normality.TsukubaJ.Math.Vol22No1(1998),213-225.[3]Beauville,A.,Complex Algebraic Surfaces,Cambridge University Press(1983).[4]Griﬃths,P.,and Harris,J.,Principles of Algebraic Geometry, A.Wiley-Interscience publication,(1978)[5]Green,M.and Lazarsfeld,R.,On the projective normality of complete linear serieson an algebraic curve,Invent.Math.83(1986),73–90.[6]Hartshorne,R.,Algebraic geometry,Graduate Text in Math,52,Berlin-Heidelberg-New York1977.[7]Kim,S.and Kim,Y.,Projectively normal embedding of a k-gonal curve,Commu-nications in Algebra32(1)187-201(2004).[8]Kim,S.and Kim,Y.,Normal generation of line bundles on algebraic curves,Jour-nal of Pure and Applied Algebra192(3)173-186(2004).[9]Livorini,E.L.,Classiﬁcation of algebraic non-ruled surfaces with sectional genusless than or equal to six,Nagoya Math.J.100,1-9(1985).[10]nge and G.Martens,Normal generation and presentation of line bundles oflow degree on curves.J.reine angew.Math.356(1985),1-18.[11]Mumford,D.,Varieties deﬁned by quadric equations,Corso C.I.M.E.1969,inQuestions on Algebraic Varieties,Cremonese,Rome83(1970),30–100.[12]Sommese, A.J.,Hyperplane sections of projective surfaces.I,The adjunctionmapping,Duke Math.J.46,377-401(1979)7 Seonja Kim,Department of Electronics,Chungwoon University,Chungnam,350-701,Ko-reaE-mail address:sjkim@chungwoon.ac.krYoung Rock Kim,Department of Mathematics,Konkuk University,Seoul,143-701,Korea E-mail address:rocky777@math.snu.ac.kr。