Exponential map - Wikipedia, the free encyclopedia
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography.
Exponential map
From Wikipedia, the free encyclopedia
In differential geometry, the exponential map is a
generalization of the ordinary exponential function
of mathematical analysis to all differentiable
manifolds with an affine connection. Two important
special cases of this are the exponential map for a
manifold with a Riemannian metric, and the
exponential map from a Lie algebra to a Lie group.
Contents
1 Definition
2 Lie theory
2.1 Definitions
2.2 Examples
2.3 Properties
3 Riemannian geometry
3.1 Properties
4 Relationships
5 See also
6 Notes
7 References Definition
Let M be a differentiable manifold and p a point of M . An affine connection on M allows one to define the notion of a geodesic through the point p .[1]
Let v ∈ T p M be a tangent vector to the manifold at p . Then there is a unique
geodesic γv satisfying γv (0) = p with initial tangent vector γ′v (0) = v . The corresponding exponential map is defined by exp p (v ) = γv (1). In general, the
exponential map is only locally defined , that is, it only takes a small neighborhood of the origin at T p M , to a neighborhood of p in the manifold. This is because it relies on the theorem on existence and uniqueness for ordinary differential
equations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
Lie theory
In the theory of Lie groups, the exponential map is a map from the Lie algebra of
a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
Definitions
Let be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of ). The exponential map is a map
which can be defined in several different ways as follows:
It is the exponential map of a canonical left-invariant affine connection on G, such that parallel transport is given by left translation.
It is the exponential map of a canonical right-invariant affine connection on
G. This is usually different from the canonical left-invariant connection, but
both connections have the same geodesics (orbits of 1-parameter subgroups
acting by left or right multiplication) so give the same exponential map.
It is given by where
is the unique one-parameter subgroup of whose tangent vector at the identity is equal to . It follows easily from the chain rule that .
The map may be constructed as the integral curve of either the right- or
left-invariant vector field associated with . That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
If is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:
(here is the identity matrix).
If G is compact, it has a Riemannian metric invariant under left and right
translations, and the exponential map is the exponential map of this Riemannian
metric.
Examples
The unit circle centered at 0 in the complex plane is a Lie group (called the circle group) whose tangent space at 1 can be identified with the imaginary line in the complex plane, The exponential map for this Lie group is given by
that is, the same formula as the ordinary complex exponential.
In the split-complex number plane the imaginary line forms the Lie algebra of the unit hyperbola group
since the exponential map is given by
The unit 3-sphere centered at 0 in the quaternions H is a Lie group
(isomorphic to the special unitary group ) whose tangent space at 1 can be identified with the space of purely imaginary quaternions,
The exponential map for this Lie group is given by
This map takes the 2-sphere of radius inside the purely imaginary
quaternions to a 2-sphere of radius
when . Compare this to the first example above.
Properties
For all , the map is the unique one-parameter subgroup of whose tangent vector at the identity is . It follows that:
The exponential map is a smooth map. Its derivative at the
identity, , is the identity map (with the usual identifications).
The exponential map, therefore, restricts to a diffeomorphism from some
neighborhood of 0 in to a neighborhood of 1 in .
The exponential map is not, however, a covering map in general – it is not a
local diffeomorphism at all points. For example, so(3) to SO(3) is not a
covering map; see also cut locus on this failure.
The image of the exponential map always lies in the identity component of .
When is compact, the exponential map is surjective onto the identity
component.
The image of the exponential map of the connected but non-compact group SL2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with module 1, and of non-diagonalizable
trigonalizable matrices with eigenvalue 1.
The map is the integral curve through the identity of both the right- and left-invariant vector fields associated to .
The integral curve through of the left-invariant vector field
associated to is given by . Likewise, the integral curve through of the right-invariant vector field is given by . It follows that the flows generated by the vector fields are given by:
Since these flows are globally defined, every left- and right-invariant vector field on is complete.
Let be a Lie group homomorphism and let be its derivative at the
identity. Then the following diagram commutes:
In particular, when applied to the adjoint action of a group we have
Riemannian geometry
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.
Properties
Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and going in that direction, for a unit time. Since v corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define
exp p(v) = β(|v|) where β is the unit-speed geodesic (geodesic parameterized by arc length) going in the direction of v. As we vary the tangent vector v we will get, when applying exp p, different points on M which are within some distance from the base point p—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold.
The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space (which justifies the usual term geodesically complete for a manifold having an exponential map with this property). In particular, compact manifolds are geodesically complete. However even if exp p is defined on the whole tangent space,
it will in general not be a global diffeomorphism. However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of T p M on which the exponential map is an embedding (i.e., the exponential map is a local diffeomorphism). The radius of the largest ball about the origin in T p M that can be mapped diffeomorphically via exp p is called the injectivity radius of M at p. The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
An important property of the exponential map is the following lemma of Gauss (yet another Gauss's lemma): given any tangent vector v in the domain of definition of exp p, and another vector w based at the tip of v (hence w is actually in the double-tangent space T v(T p M)) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map. This means, in particular, that the boundary sphere of a small ball about the origin in T p M is orthogonal to the geodesics in M determined by those vectors (i.e., the geodesics are radial). This motivates the definition of geodesic normal coordinates on a Riemannian manifold.
The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration. Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through p determined by the image under exp p of a 2-dimensional subspace of T p M.
Relationships
In the case of Lie groups with a bi-invariant metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-
Riemannian structure are the same as the exponential maps of the Lie group. In general, Lie groups do not have a bi-invariant metric, though all connected semi-simple (or reductive) Lie groups do. The existence of a bi-invariant Riemannian metric is stronger than that of a pseudo-Riemannian metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact (or abelian) Lie group has such a Riemannian metric.
Take the example that gives the "honest" exponential map. Consider the positive real numbers R+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product
(multiplying them as usual real numbers but scaling by y2). (This is what makes the metric left-invariant, for left multiplication by a factor will just pull out of the inner product, twice — canceling the square in the denominator).
Consider the point 1 ∈ R+, and x ∈ R an element of the tangent space at 1. The usual straight line emanating from 1, namely y(t) = 1 + xt covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed ("constant speed", remember, is not going to be the ordinary constant speed, because we're using this funny metric). To do this we reparametrize by arc length (the integral of the length of the tangent vector in the norm induced by the modified metric):
and after inverting the function to obtain t as a function of s, we substitute and get
Now using the unit speed definition, we have
,
giving the expected e x.
The Riemannian distance defined by this is simply
,
a metric which should be familiar to anyone who has drawn graphs on log paper. See also
List of exponential topics
Notes
1. ^ A source for this section is Kobayashi & Nomizu (1975, §III.6), which uses the term
"linear connection" where we use "affine connection" instead.
References
do Carmo, Manfredo P. (1992), Riemannian Geometry, Birkh?user, ISBN 0-8176-3490-8. See Chapter 3.
Cheeger, Jeff; Ebin, David G. (1975), Comparison Theorems in Riemannian
Geometry, Elsevier. See Chapter 1, Sections 2 and 3.
Hazewinkel, Michiel, ed. (2001), "Exponential mapping"
(https://www.sodocs.net/doc/98200548.html,/index.php?title=p/e036930), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, Providence, R.I.: American
Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454
(https://https://www.sodocs.net/doc/98200548.html,/mathscinet-getitem?mr=1834454).
Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential
Geometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3.
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博客的发展及演变
博客的发展及 演变 学院：历史文化学院 专业：文化产业管理 班级：1201班 姓名：聂康康 学号：1205024107
摘要：无论是在国外还是国内，博客和博客文化正以“润物细无声”的方式深刻影响着人们的生活。博客是个人性和公共性的结合体。博客精神的核心并不是自娱自乐，也不仅是个人表达自由，准确地说，博客体现的是一种利他的共享精神，为他人提供帮助。个人日记和个人网站主要表现的还是“小我”，而博客表现的是“大我”。两者也许形式上很接近，但内在有着本质的差异。 关键词：博客、互联网 博客已经成为互联网文化不可分割的一部分。 如今网民几乎都会阅读博客，无论是传统新闻媒介的“官方”新闻博客、与自己爱好兴趣相关的话题性博客或是纯娱乐博客，几乎每一个人都会有一两个特别钟爱的博客。 但以前并不是这样。与互联网本身相比，博客的历史并不算长。博客真正兴起并成为互联网风景的重要组成，是在最近五到十年。 美国人工智能专家乔恩·巴杰（Jorn Barger）1997年12月在其网站上首次使用了weblog一词；2002年，博客开始引入中国，数量不足1万人；2002年7月，blog的中文“博客”由方兴东、王俊秀正式命名；2002年8月，方兴东、王俊秀开通博客中国（blogchina）
网站；2004年以来，博客主页（weblog或blog）——一种采用简便的软件生成个人主页、能够按照时间顺序不断更新、实现个人信息的历时积累和传播的互联网个人出版方式，在我国进入迅猛发展时期。 博客（blogger）概念解释为网络出版（Web Publishing）、发表和张贴（Post-这个字当名词用时就是指张贴的文章）文章，是个急速成长的网络活动，现在甚至出现了一个用来指称这种网络出版和发表文章的专有名词——Weblog或Blog。 Blogger即指撰写Blog的人。Blogger在很多时候也被翻译成为“博客”一词，而撰写Blog这种行为，有时候也被翻译成“博客”。因而，中文“博客”一词，既可作为名词，分别指代两种意思Blog （网志）和Blogger（撰写网志的人），也可作为动词，意思为撰写网志这种行为，只是在不同的场合分别表示不同的意思罢了。 Blog是一个网页，通常由简短且经常更新的帖子（Post，作为动词，表示张贴的意思，作为名字，指张贴的文章）构成，这些帖子一般是按照年份和日期倒序排列的。而作为Blog的内容，它可以是你纯粹个人的想法和心得，包括你对时事新闻、国家大事的个人看法，或者你对一日三餐、服饰打扮的精心料理等，也可以是在基于某一主题的情况下或是在某一共同领域内由一群人集体创作的内容。它并不等同于“网络日记”。作为网络日记是带有很明显的私人性质的，而Blog则是私人性和公共性的有效结合，它绝不仅仅是纯粹个人思想的表达和日常琐事的记录，它所提供的内容可以用来进行交流和为他人提供帮助，是可以包容整个互联网的，具有极高的共享精神和价值。
未来5年所有行业的发展趋势
未来5年所有行业的发展趋势 核心：预测未来5年所有16个行业的发展趋势！互联网最有价值的不是自己在产生很多新东西，而是对已有行业的潜力再次挖掘，用互联网的思维去重新提升传统行业。 那么从这个角度去观察，互联网影响传统行业的特点有几点： 第一，打破信息的不对称性格局，竭尽所能透明一切信息。 第二，对产生的大数据进行整合利用，使得资源利用最大化。 第三，互联网的群蜂意志拥有自我调节机制。我把人类群体思维模式称之为群蜂意志，你可以想象一个人类群体大脑记忆库的建立：最初的时候各个神经记忆节点的搜索路径是尚未建立的，当我们需要反复使用的时候就慢慢形成强的连接，在互联网诞生之前这些连接记忆节点的路径是微弱的，强连接是极少的，但是互联网出现之后这些路径瞬间全部亮起，所有记忆节点都可以在瞬间连接。这样就给了人类做整体未来决策有了超越以往的前所未有的体系支撑，基于这样的记忆模式，人类将重新改写各个行业，以及人类未来。 以下是对各行业的盘点，涉及面较多，有些部分笔者观察尚浅，还望多包涵。 1.零售业 传统零售业对于消费者来说最大的弊端在于信息的不对称性。在《无价》
一书中，心理实验表明外行人员对于某个行业的产品定价是心里根本没有底的，只需要抛出锚定价格，消费者就会被乖乖的牵着鼻子走。而C2C,B2C却完全打破这样的格局，将世界变平坦，将一件商品的真正定价变得透明。大大降低了消费者的信息获取成本。让每一个人都知道这件商品的真正价格区间，使得区域性价格垄断不再成为可能，消费者不再蒙在鼓里。 不仅如此，电子商务还制造了大量用户评论UGC。这些UGC真正意义上制造了互联网的信任机制。而这种良性循环，是传统零售业不可能拥有的优势。 预测未来的零售业, 第一，会变成线下与线上的结合，价格同步。 第二，同质化的强调功能性的产品将越来越没有竞争力，而那些拥有一流用户体验的产品会脱引而出。第三，配合互联网大数据，将进行个性化整合推送(现在亚马逊就已经将首页改版为个性化推送主页)。 2.批发业 传统批发业有极大的地域限制，一个想在北京开家小礼品店的店主需要大老远的跑到浙江去进货，不仅要面对长途跋涉并且还需要面对信任问题。所以对于进货者来说，每次批发实际上都是一次风险。当阿里的B2B 出现之后，这种风险被降到最低。一方面，小店主不需要长途跋涉去亲自检查货品，只需要让对方邮递样品即可。另一方面，阿里建立的信任问责制度，使得信任的建立不需要数次的见面才能对此人有很可靠的把
谷歌发展史
谷歌发展史 1，谷歌上线 公司成立肯定是谷歌搜索历史上最重大的里程碑事件。1997年到1998年间，谷歌联合创始人拉里·佩奇（Larry Page）和谢尔盖·布林（Sergey Brin）开始在美国加州门罗帕克的一间车库内筹备公司。成立数天后，公司注册了https://www.sodocs.net/doc/98200548.html,域名。这项服务背后的概念，也就是“无穷大”（googol）这个单词，显示出公司要用一种建设性的方式，组织万维网上无穷的信息，从而帮助用户找需要的答案。 2，在雅虎的帮助下，踏出成功的第一步
2000年之前，谷歌尚未成为搜索行业的主流。行业领头羊的地位属于1994年成立的老牌搜索引擎雅虎。也正是这家公司用一笔搜索合作交易让谷歌崭露头角——雅虎放弃了Inktomi，通过布林和佩奇的服务支持自己的原生搜索结果。真正有意思的是这桩交易15年后的结果。 两家公司最终出现了很多纠纷，其中就涉及到技术专利。2004年时，谷歌与雅虎和解了了专利纠纷，通过雅虎子公司Overture Services，谷歌向雅虎发行了270万股A类普通股。谷歌的广告产品AdWords 由此建立。 3，自助式的盈利搜索模式
在过去，Lycos AskJeeves、Excite等搜索引擎通过直显广告获得收入。这种广告模式在90年代末非常受欢迎，但一段时间后便没人再点击了。搜索公司和广告主遇到了收入难题。 谷歌在2000年推出了AdWords产品。这个产品的模式在当时看起来十分新奇——品牌不用再和广告机构沟通广告上的事宜，而可以自行管理，节省了时间、精力与成本。 更有吸引力的是，AdWords中有一项个性化的尝试，广告可以按照搜索查询的内容创建，而不再是随机出现。此外，谷歌还使用竞价模式，提高了收入：出价越高，顾客的文字广告的位置就越靠上。 雅虎与微软马上效仿这种做法。这种广告在谷歌收入中占比非常大——在2012年广告总营收中是425亿美元。 4，微软加雅虎大于谷歌？ 由于对谷歌在搜索引擎的统治地位感到不满，雅虎和微软在2010年形成联盟，用微软的技术负责雅虎的搜索算法和付费搜索平台。同时，雅虎也变成了两家公司的“独家合作销售”，向两家公司的付费搜索广告主出售广告。 这桩交易影响巨大，需要得到美欧监管机构的批准，而这两地的监管者最终在2010年初放行了这桩交易。合作的效果究竟如何并不是很清楚，但雅虎在2012年4月1日起将合作延长了12个月。不过，当玛丽莎·梅耶尔（Marissa Mayer）执掌雅虎后，她曾表示，这份协议并没有带来承诺的市场份额与营收。 雅虎在今年早些时候与谷歌签订了“全球性的、非排他的内容关联广告交易”，证明它还是更喜欢谷歌多一点，也表明微软-雅虎合作正式终结。 对不起了，微软。 5，我的网站为什么排名低？
BBS的发展史
什么是BBS？ BBS的英文全称是Bulletin Board System，翻译为中文就是“电子公告板”。BBS最早是用来公布股市价格等类信息的，当时BBS连文件传输的功能都没有，而且只能在苹果计算机上运行。早期的BBS与一般街头和校园内的公告板性质相同，只不过是通过电脑来传播或获得消息而已。一直到个人计算机开始普及之后，有些人尝试将苹果计算机上的BBS转移到个人计算机上，BBS才开始渐渐普及开来。近些年来，由于爱好者们的努力，BBS的功能得到了很大的扩充。 目前，通过BBS系统可随时取得各种最新的信息；也可以通过BBS系统来和别人讨论计算机软件、硬件、Internet、多媒体、程序设计以及生物学、医学等等各种有趣的话题；还可以利用BBS系统来发布一些“征友”、“廉价转让”、“招聘人才”及“求职应聘”等启事；更可以召集亲朋好友到聊天室内高谈阔论……这个精彩的天地就在你我的身旁，只要您在一台可以访问校园网的计算机旁，就可以进入这个交流平台，来享用它的种种服务BBS 维基百科，自由的百科全书 BBS是电子公告板系统(Bulletin Board System)之英文缩写，它通过在计算机上运行服务软件，允许用户使用终端程序通过电话调制解调器拨号或者Internet来进行连接，执行下载数据或程序、上传数据、阅读新闻、与其它用户交换消息等功能。许多BBS由站长(通常被称为SYSOP-SYStem OPerator)业余维护，而另一些则提供收费服务。 目前，有的时候BBS也泛指网络论坛或网络社群。 目录 [隐藏] * 1 BBS技术及常见软件 * 2 BBS人文文化 o 2.1 中国大陆BBS“系统维护”现象 o 2.2 BBS用语 * 3 参看 * 4 外部链接 [编辑] BBS技术及常见软件 因特网（Internet）之前，在20世纪80年代中叶就开始出现基于调制解调器（modem）和电话线通信的拨号BBS及其相互连接而成的BBS网络。 后来随着因特网的普及，拨号BBS和BBS网络已经日渐凋零，所剩无几。目前的BBS站点，多数是基于Internet的Telnet协议。在服务器端，采用Maple BBS或者FireBird BBS 系统。用户端通过Telnet软件如NetTerm、CTerm、FTerm等来登陆服务器，阅读发表文
搜索引擎发展历史
搜索引擎成为互联网的重要应用之一 ??? 从90年代末开始，互联网上的网站与网页数量飞速增长，网民的兴趣点也从屈指可数的几家综合门户类网站分散到特色各异的中小网站去了。人们想在互联网上找到五花八门的信息，但由于人工分类编辑网站目录的方法受到时效和收录量的限制，无法再满足人们对网上内容的检索需求，于是搜索引擎在2000年后开始大行其道。使用蜘蛛程序在互联网上自动抓取海量网页信息，索引并存储到庞大的数据库中，并通过特殊算法将相关性最好的结果瞬间呈现给搜索者，搜索引擎的便捷使其成为互联网最受欢迎的应用之一。以至于有相当多的人将浏览器的默认首页设为搜索引擎，甚至形成了将网站名称输入到搜索框中而非浏览器地址栏这样独特的网络导航习惯。 呼叫目录返回顶部 搜索成为人们思考行为的一部分 ??? 随着网上社区（SNS），博客（Blog），维基百科（Wikipedia）等如火如荼的发展，网民从单纯的信息获取者演变成信息发布者，人们通过网络分享自己的知识、体验、情感或见闻，使互联网上的内容越来越丰富多彩。例如，按照统计，目前中国网民在百度知道平台上的问题解决率高达97.9%，这些问题涉及科技、社会、文化、商业等各个方面，尤其对人们的衣食住行等日常生活问题，几乎都能从平台获得满意的答案。截至到09年7月的4年时间内，中文互动问答平台百度知道已经累计为中国网民解决了5650多万个问题，成为人们日常生活的最佳互动问答平台。社区内容上的无所不谈使搜索引擎的收录也变得无所不包，人们发现通过搜索引擎可以找到他想要的任何信息，从新闻热点到柴米油盐，从育儿百科到MBA课程。信息的便捷获取潜移默化的改变了人们的思考行为，搜索结果页上汇集了整个互联网的智慧，谁不想在苦思冥想前“搜索一下”呢？ 呼叫目录返回顶部 搜索成为人们消费行为的重要环节 ??? 随着对搜索引擎的依赖加深，当人们有消费需求或看到感兴趣的商品时，“搜索一下”已经是已形成的“条件反射”。以前，消费者依靠“货比三家”来对抗“买的没有卖的精”这种与商家之间的信息不对称。现在，通过搜索引擎收集到的产品功能与使用情况弥补了消费者与推广商家间在知情权上的鸿沟，成为消费决策的重要依据。价格低的线上销售渠道也成为搜索热点，以至于现在出现了消费者为省钱而先到实体专卖店挑选合适型号大小货品再到网店付款下单的有趣现象。随着年轻一代消费能力的提高，从前仅限于图书音像和电子产品的网上购物正在向工作生活的各个层面迅速渗透，服装食品等日用消费品也逐渐成为网购的宠儿。 呼叫目录返回顶部 搜索营销原理 原理介绍 ????传统营销需要选择目标市场，通过创造、传递、传播优质的客户价值，获得、保持和发展优质客户。而在互联网时代，网站由于其内容丰富、查阅方便、不受时空限制、成本低等优势，广受网民和商家的喜爱，成为传递、传播价值的主要手段，并在获得、保持和发展客户方面呈现强大的潜力。所以，围绕网站的营销活动越来越丰富。