仿射空间.docx

詹森不等式以丹麦数学家约翰詹森（Johan Jensen）命名。它给出积分的凸函数值和凸函数的积分值间的关系。 Jensens inequality generalizes the statement that a secant line of a convex function lies above the graph.詹森不等式的一般形式詹森不等式可以用测度论或概率论的语言给出。这两种方式都表明同一个很一般的结果。 测度论的版本假设是集合的正测度，使得() = 1。若g是勒贝格可积的实值函数，而是在g的值域上定义的凸函数，则 。 概率论的版本以概率论的名词，是个概率测度。函数g换作实值随机变量X（就纯数学而言，两者没有分别）。在空间上，任何函数相对于概率测度的积分就成了期望值。这不等式就说，若是任一凸函数，则 。 詹森不等式的特例机率密度函数的形式假设是实数轴上的可测子集，而f(x)是非负函数，使得 。 以概率论的语言，f是个机率密度函数。 詹森不等式变成以下关于凸积分的命题： 若g是任一实值可测函数，在g的值域中是凸函数，则 。 若g(x) = x，则这形式的不等式简化成一个常用特例： 。 有限形式若是有限集合，而是上的正规计数测度，则不等式的一般形式可以简单地用和式表示： ， 其中。 若是凹函数，只需把不等式符号调转。 假设是正实数，g(x) = x，i = 1 / n及。上述和式便成了 ， 两边取自然指数就得出熟悉的平均数不等式： 。 这不等式也有无限项的离散形式。 统计物理学统计物理学中，若凸函数是指数函数，詹森不等式特别重要： ， 其中方括号表示期望值，是以随机变量X的某个概率分布算出。这个情形的证明很简单（参见Chandler, Sec. 5.5）：在以下等式的第三个指数函数 套用不等式 ， 詹森不等式(Jensensinequity)证明方法备忘录詹森不等式(Jensens inequity)证明方法备忘录若f为凹函数，即f=0，且有a1+a2+a3+a4+an=1成立，则：f(a1*x1+a2*x2+a3*x3+an*xn)=a1*f(x1)+a2*f(x2)+an*xn恒成立。证明：不是一般性，令xi=x(i+1),(1)首先证明当n=2时，fa1*x1+(1-a1)*x2=a1*f(x1)+(1-a1)*f(x2)成立欲证上式成立，即证明a1+(1-a1) *fa1*x1+(1-a1)*x2 =a1*f(x1)+(1-a1)*f(x2)成立，移项并合并同类项后上式可变为：a1*fa1*x1+(1-a1)*x2- f(x1)=(1-a1)* f(x2)- fa1*x1+(1-a1)*x2（1）根据罗尔中值定理，有：f(x)-f(x)/(x-x)=f(y),x=y=x，因此(1)式可变为：a1*f(y1)*(a1-1)*(x1-x2)= (1-a1)*f(y1)*a1*(x2-x1)=(1-a1)*f(y2)*a1*(x2-x1),其中y2=y1。由f=0可得，f(y1)=f(y2)，因此（1）式成立。（注：麦克劳林展开只有在（x2-x1)趋于零时成立，因此，此处不能使用麦克劳林展开公式）。（2）证明n=2k(k为正整数)时命题成立，首先证明n=4时原命题成立，则由（1）可得：（a1+a2）*fa1/(a1+a2)*x1+a2/(a1+a2)*x2=(a1+a2)*a1/(a1+a2)f(x1)+a2/(a1+a2)*f(x2)(2)（a3+a4）*fa3/(a3+a4)*x3+a4/(a3+a4)*x4=(a3+a4)*a3/(a3+a4)f(x3)+a4/(a3+a4)*f(x4)(3)将上述（2）式与（3）式同时除以（a1+a2+a3+a4），再次利用（1）式可得：fa1/(a1+a2+a3+a4)*x1+a2/( a1+a2+a3+a4)*x2+a3/( a1+a2+a3+a4)*x3+ a4/( a1+a2+a3+a4)*x4=(a1+a2)/(a1+a2+a3+a4)* fa1/(a1+a2)*x1+a2/(a1+a2)*x2+（a3+a4）/(a1+a2+a3+a4)*fa3/(a3+a4)*x3+a4/(a3+a4)*x4(4)由于a1+a2+a3+a4=1，因此（4）式左边部分即为f(a1*x1+a2*x2+a3*x3+a4*x4)，右边部分即为a1*f(x1)+a2*f(x2)+a3*f(x3)+a4*f(x4)，n=4时，命题得证。同理可从最底层开始运用公式（1）证明n=2k(k1且k2)的情形依然成立。（3）当n2k时，可将ai*xi分成m个部分，即m个ai/m*xi之和，使得n+(m-1)=2k，再利用上式便可直接得到原命题。（4）当ai均趋向于0时，取其极限形式，便可证明fE(x)=Ef(x)，将f函数符号改为U符号，即得微观经济学中冯诺依曼期望效用的一个不等式。DefinitionAnaffine spaceis a settogether with avector spaceand agroup actionof(with addition of vectors as group operation) on, such that the only vector acting with a fixpoint is(i.e., the action is free) and there is a single orbit (the action is transitive). In other words, an affine space is aprincipal homogeneous spaceover the additive group of a vector space.Explicitly, an affine space is a point settogether with a mapwith the following properties:1. Left identity2. Associativity3. Uniquenessis abijection.Thevector spaceis said to underlie theaffine spaceand is also called thedifference space.By choosing an origin, one can thus identifywith, hence turninto a vector space. Conversely, any vector space, is an affine space over itself. Theuniquenessproperty ensures that subtraction of any two elements ofis well defined, producing a vector of.If, andare points inandis a scalar, thenis independent of. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.By noting that one can define subtraction of points of an affine space as follows:is the unique vector insuch that,one can equivalently define an affine space as a point set, together with a vector space, and a subtraction mapwith the following properties2:1. there is a unique pointsuch thatand2. .These two properties are calledWeyls axioms.editExamples When children find the answers to sums such as 4+3 or 42 by counting right or left on anumber line, they are treating the number line as a one-dimensional affine space. Anycosetof a subspaceof a vector space is an affine space over. Ifis a matrix andlies in its column space, the set of solutions of the equationis an affine space over the subspace of solutions of. The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, ifis a linear mapping andylies in its image, the set of solutionsto the equationis a coset of the kernel of, and is therefore an affine space over.editAffine subspacesAnaffine subspace(sometimes called alinear manifold,linear variety, or aflat) of a vector spaceis a subset closed under affine combinations of vectors in the space. For example, the setis an affine space, whereis a family of vectors in this space is theaffine spanof these points. To see that this is indeed an affine space, observe that this set carries a transitive action of thevector subspaceofThis affine subspace can be equivalently described as the coset of the-actionwhereis any element of, or equivalently as anylevel setof thequotient mapA choice ofgives a base point ofand an identification ofwithbut there is no natural choice, nor a natural identification ofwithA linear transformation is a function that preserves alllinear combinations; an affine transformation is a function that preserves allaffine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.For example, in, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.editAffine combinations and affine dependenceMain article:Affine combinationAnaffine combinationis a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors arelinearly independentif none is a linear combination of the others, so also they areaffinely independentif none is an affine combination of the others. The set of linear combinations of a set of vectors is their linear span and is always a linear subspace; the set of all affine combinations is their affine span and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of threenon-collinear pointsis the plane that contains all three.Vectorsv1,v2, .,vnare linearly dependent if there exist scalarsa1,a2, ,an, not all zero, for whicha1v1+a2v2+ +anvn=0(1)Similarly they areaffinely dependentif in addition the sum of coefficients is zero:editAxiomsAffine space is usually studied asanalytic geometryusing coordinates, or equivalently vector spaces. It can also be studied assynthetic geometryby writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.Coxeter (1969, p.192) axiomatizes affine geometry (over the reals) asordered geometrytogether with an affine form ofDesarguess theoremand an axiom stating that in a plane there is at most one line through a given point not meeting a given line.Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint): Any two distinct points lie on a unique line. Given a point and line there is a unique line which contains the point and is parallel to the line There exist three non-collinear points.As well as affine planes over fields (ordivision rings), there are also manynon-Desarguesian planessatisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher dimensional affine spaces.editRelation to projective spacesAn affine space is a subspace of projective space, which is in turn a quotient of a vector space.Affine spaces aresubspacesofprojective spaces: an affine plane can be obtained from anyprojective planeby removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as aclosureby adding a line at infinity whose points correspond to equivalence classes of parallel lines.Further, transformations of projective space that preserve affine space (equivalently, that preserve the points at