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## 数学代写|黎曼几何代写Riemannian geometry代考|Bimodule Quantum Levi-Civita Connections

Let $(A, \Omega$, d) be an exterior algebra in the sense of Chap. 1 , specified at least to degree 2 . We have already met the notion of a connection on a general $A$-module in Chap. 3 but now we focus exclusively on connections $\nabla: \Omega^1 \rightarrow \Omega^1 \otimes_A \Omega^1$ on $\Omega^1$. As usual, a left connection obeys a left Leibniz rule
$$\nabla(a \eta)=\mathrm{d} a \otimes \eta+a \nabla \eta, \quad a \in A, \eta \in \Omega^1$$
and has curvature $R_{\nabla}$ and torsion $T_{\nabla}$ given by left $A$-module maps
$$\begin{gathered} R_{\nabla}: \Omega^1 \rightarrow \Omega^2 \otimes_A \Omega^1, \quad R_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) \nabla \ T_{\nabla}: \Omega^1 \rightarrow \Omega^2, \quad T_{\nabla}=\wedge \nabla-\mathrm{d} \end{gathered}$$
where $\wedge: \Omega^1 \otimes_A \Omega^1 \rightarrow \Omega^2$ is the exterior product. We have met both formulae before in Chap. 3 and then again in Chap. 5. The concept of a connection itself requires only $\Omega^1$, while curvature and torsion require $\Omega^2$ in the role, classically, of defining curvature and torsion on antisymmetric combinations of vector fields. In Chap. 5 , coming from quantum frame bundles, we were also led to introduce a new tensor built from a metric and a connection, the cotorsion, defined as
$$\operatorname{co} T_{\nabla} \in \Omega^2 \otimes_A \Omega^1, \quad \operatorname{co} T_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) g$$

## 数学代写|黎曼几何代写Riemannian geometry代考|More Examples of Bimodule Riemannian Geometries

Here we use bases of 1-forms to write explicit formulae for ideas which we previously discussed in a basis-free fashion. The existence of the basis $\left{e^i\right}$ corresponds to the assumption that $\Omega^1$ is finitely generated projective as a left module as in $\S 3.1$, and the uniqueness of the coefficients of the basis elements in the following formulae corresponds to $\Omega^1$ being left-parallelisable as in Definition 1.2. Without the latter we would have to insert a projection matrix in various places (this generality is discussed in Chap. 3) so to keep things simple here we proceed under the assumption that $\Omega^1$ is left-parallelisable. To fix conventions, we write basis 1-forms $e^i$ with indices $u p$, which has not been our preference in most of the book where we have tended to use lower indices where possible as upper ones clash with powers. This is needed to fit conventions in physics and we combine this with Einstein’s summation convention where repeated up-down pairs of indices are to be summed. Thus the defining formulae for ‘partial derivatives’ from Chap. 1 and left connections in terms of Christoffel symbols from $\S 3.2$ now appear as,
$$e^i a=C^i{ }j(a) e^j, \quad \mathrm{~d} a=\left(\partial_i a\right) e^i, \quad \nabla\left(e^i\right)=-\Gamma^i{ }{j k} e^j \otimes e^k$$
for all $a \in A$ in our coordinate algebra. If $e^i$ and $a$ commute (e.g. if $a$ is an element of the field $\mathbb{k}$, which we refer to loosely as a constant) then $C^i{ }j(a)=a \delta^i{ }_j$. For a bimodule connection we write $\sigma$ as $$\sigma\left(e^i \otimes e^j\right)=\sigma^{i j}{ }{m n} e^m \otimes e^n$$
with coefficients determined from $\Gamma^i{ }{j k}$ and $C^i{ }_j$ and such that $\sigma$ extends as a bimodule map, which will depend on $\Gamma^i{ }{j k}$ as not every left connection is necessarily a bimodule connection. We next suppose that there is a central metric $g=g_{i j} e^i \otimes e^j$ and define the inverse-metric tensor as $g^{i j}=\left(e^i, e^j\right)$. This is inverse in the sense that
$$g_{i j} C^i{ }n\left(g^{j k}\right)=\delta^k{ }_n, \quad C^k{ }_p\left(g{i j}\right) g^{p i}=\delta^k{ }j$$ while centrality of $g$ comes down to $$a g{i j}=g_{q s} C^q\left(C^s{ }_j(a)\right)$$

for all $a \in A$. We give one detailed calculation of converting tensor product notation to index notation and leave the rest to the reader. Namely, the equation for metric compatibility $\nabla g=0$ is
$$\begin{gathered} \mathrm{d} g_{i j} \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} \sigma\left(e^i \otimes \Gamma_{p k}^j e^p\right) \otimes e^k \ \left(\partial_r g_{i j}\right) e^r \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} C_q^i\left(\Gamma_{p k}^j\right) \sigma^{q p}{ }{r m} e^r \otimes e^m \otimes e^k \end{gathered}$$ so on re-indexing and taking coefficients of the basis elements we get the equation $$\partial_r g{m n}=g_{i n} \Gamma_{r m}^i+g_{i j} C_q^i\left(\Gamma_{p n}^j\right) \sigma_{r m}^{q p}$$

## 数学代写|黎曼几何代写riemanannian geometry代考|双模量子Levi-Civita连接

$$\nabla(a \eta)=\mathrm{d} a \otimes \eta+a \nabla \eta, \quad a \in A, \eta \in \Omega^1$$
，并具有由左$A$ -模块映射
$$\begin{gathered} R_{\nabla}: \Omega^1 \rightarrow \Omega^2 \otimes_A \Omega^1, \quad R_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) \nabla \ T_{\nabla}: \Omega^1 \rightarrow \Omega^2, \quad T_{\nabla}=\wedge \nabla-\mathrm{d} \end{gathered}$$

$$\operatorname{co} T_{\nabla} \in \Omega^2 \otimes_A \Omega^1, \quad \operatorname{co} T_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) g$$

## 数学代写|黎曼几何代写黎曼几何代考|更多双模黎曼几何的例子

$$e^i a=C^i{ }j(a) e^j, \quad \mathrm{~d} a=\left(\partial_i a\right) e^i, \quad \nabla\left(e^i\right)=-\Gamma^i{ }{j k} e^j \otimes e^k$$

，其中的系数由$\Gamma^i{ }{j k}$和$C^i{ }_j$确定，并使$\sigma$扩展为一个双模块映射，这将依赖于$\Gamma^i{ }{j k}$，因为并非每个左连接都一定是一个双模块连接。我们接下来假设有一个中心度规$g=g{i j} e^i \otimes e^j$，并定义逆度规张量$g^{i j}=\left(e^i, e^j\right)$。这与
$$g_{i j} C^i{ }n\left(g^{j k}\right)=\delta^k{ }n, \quad C^k{ }_p\left(g{i j}\right) g^{p i}=\delta^k{ }j$$相反，而$g$的中心性可归结为$$a g{i j}=g{q s} C^q\left(C^s{ }_j(a)\right)$$

for all $a \in A$。我们给出了一个将张量积表示法转换为索引表示法的详细计算，其余的留给读者。也就是说，度规兼容性的方程$\nabla g=0$是
$$\begin{gathered} \mathrm{d} g_{i j} \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} \sigma\left(e^i \otimes \Gamma_{p k}^j e^p\right) \otimes e^k \ \left(\partial_r g_{i j}\right) e^r \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} C_q^i\left(\Gamma_{p k}^j\right) \sigma^{q p}{ }{r m} e^r \otimes e^m \otimes e^k \end{gathered}$$，所以重新索引并取基本元素的系数，我们得到方程$$\partial_r g{m n}=g_{i n} \Gamma_{r m}^i+g_{i j} C_q^i\left(\Gamma_{p n}^j\right) \sigma_{r m}^{q p}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

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## 数学代写|黎曼几何代写Riemannian geometry代考|Complex Structures and the Dolbeault Double Complex

We start by reviewing the case of a classical complex manifold, where we can work locally in open subsets of $\mathbb{C}^m$ and use holomorphic maps to change between local coordinates. The most well-known nontrivial example is the Riemann sphere $\mathbb{C P}^1$ or $\mathbb{C}_{\infty}$. This has two open charts, $U=\mathbb{C}$ and $V=\mathbb{C}$, and the change of coordinate map is such that $z \in U \backslash{0}$ corresponds to $z^{-1} \in V \backslash{0}$. We shall write the local complex coordinates as $z^1, \ldots, z^m$ for a complex manifold $M$ of dimension $m$. Then $M$ also has the structure of a real $2 m$-dimensional manifold, as we have $z^j=x^j+\mathrm{i} y^j$ for real coordinates $x^1, \ldots, x^m, y^1, \ldots, y^m$.

Locally the complex-valued 1 -forms on $M$ have basis $\mathrm{d} z^j=\mathrm{d} x^j+\mathrm{id} y^j$ and its conjugate $\mathrm{d} \bar{z}^j=\mathrm{d} x^j-\operatorname{id} y^j$. Dually,
$$\frac{\partial}{\partial z^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}-\mathrm{i} \frac{\partial}{\partial y^j}\right), \quad \frac{\partial}{\partial \bar{z}^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}+\mathrm{i} \frac{\partial}{\partial y^j}\right)$$
for vector fields. It is common to abbreviate $\partial_j=\frac{\partial}{\partial z^j}$ and $\bar{\partial}_j=\frac{\partial}{\partial \bar{z}}$. The CauchyRiemann equations are the condition for a complex-valued function of a complex variable to be analytic. They are usually written in terms of splitting the function into real parts $u+\mathrm{i} v$ and the single complex variable $z=x+\mathrm{i} y$ as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$. These can be written using the $\bar{\partial}$ operators in several complex variables for a complex-valued function $f$ on $M$ as $\bar{\partial}_j f=0$ for all $j$. By writing the $n$ forms in terms of $\mathrm{d} z^j$ and $\mathrm{d} \bar{z}^j$, we split $\Omega^n(M)$ into the direct sum of $\Omega^{p, q}(M)$ for $p+q=n$. Here $p$ is the number of $\mathrm{d} z^j$ and $q$ is the number of $\mathrm{d} \bar{z}^j$, so $\mathrm{d} z^1 \wedge \mathrm{d} z^2 \wedge \mathrm{d} z^3$ is in $\Omega^{3,0}(M)$ and $\mathrm{d}^{-1} \wedge \mathrm{d} z^2 \wedge \mathrm{d}^{-3}$ is in $\Omega^{1,2}(M)$. Now define derivatives $\partial$ : $\Omega^{p, q}(M) \rightarrow \Omega^{p+1, q}(M)$ and $\bar{\partial}: \Omega^{p, q}(M) \rightarrow \Omega^{p, q+1}(M)$ by
$$\partial \xi=\mathrm{d} z^j \wedge \partial_j \xi, \quad \bar{\partial} \xi=\mathrm{d} \bar{z}^j \wedge \bar{\partial}_j \xi$$
summed over $j$. It is fairly easy to check that $\partial+\bar{\partial}=\mathrm{d}$ and that $\partial^2=\bar{\partial}^2=0$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Holomorphic Modules and Dolbeault Cohomology

Throughout this section, $(\Omega, \mathrm{d}, *, J)$ is an integrable almost complex structure on $A$. The noncommutative equivalent of the classical Cauchy-Riemann condition for $a \in A$ to be holomorphic is that $\bar{\partial} a=0$, and we call the collection of such holomorphic elements $A_{\mathrm{hol}}$. As $\bar{\partial}$ is a derivation, $A_{\mathrm{hol}}$ is a subalgebra of $A$. Similarly define $\Omega_{\text {hol }}^p$, the holomorphic $p$-forms, as the elements of $\xi \in \Omega^{p, 0}$ for which $\bar{\partial} \xi=0$. The holomorphic forms form a sub-DGA of the de Rham complex, the holomorphic de Rham complex,
$$0 \longrightarrow A_{\mathrm{hol}} \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^1 \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^2 \stackrel{\partial}{\longrightarrow} \cdots$$
For a complex analytic manifold $M$ it is natural to consider vector bundles with fibre $\mathbb{C}^n$ where the transition functions are holomorphic. In real differential geometry, the ‘obvious’ way to differentiate sections of a vector bundle is to take a trivialising open set, and then simply apply partial derivative $\frac{\partial}{\partial x^i}$ to the components of the section. Of course, this does not work globally as the derivatives of the transition functions enter, forcing us to use covariant derivatives and Christoffel symbols. However, in complex differential geometry, the ‘obvious’ thing to do works, with one condition. If we take the $\bar{\partial}$ derivatives of the components of the section, we get $\bar{\partial}_E: E \rightarrow \Omega^{0,1} \otimes_A E$ defined locally by
$$\bar{\partial}_E(v)=\mathrm{d} \bar{z}^i \otimes \frac{\partial v^j}{\partial \bar{z}^i} e_j,$$
where $e_j$ is the local basis of the vector bundle, $E$ is the sections of the bundle and $v=v^j e_j \in E$. Furthermore, this formula is perfectly well behaved under holomorphic change of basis as the $\bar{\partial}$ derivatives of the transition functions are zero, so we get a globally defined derivative. Thus, every complex vector bundle with holomorphic transition functions (we will just say holomorphic vector bundle) has a well-defined operator $\bar{\partial}_E$ satisfying the left $\bar{\partial}$-Leibniz rule, for $v \in E$ and $a \in A$,
$$\partial_E(a \cdot v)=\bar{\partial} a \otimes v+a \cdot \partial_E(v) .$$

## 数学代写|黎曼几何代写黎曼几何代考|复结构和Dolbeault双复复数

$$\frac{\partial}{\partial z^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}-\mathrm{i} \frac{\partial}{\partial y^j}\right), \quad \frac{\partial}{\partial \bar{z}^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}+\mathrm{i} \frac{\partial}{\partial y^j}\right)$$

$$\partial \xi=\mathrm{d} z^j \wedge \partial_j \xi, \quad \bar{\partial} \xi=\mathrm{d} \bar{z}^j \wedge \bar{\partial}_j \xi$$

## 数学代写|黎曼几何代写黎曼几何代考|全纯模和Dolbeault上同调

$$0 \longrightarrow A_{\mathrm{hol}} \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^1 \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^2 \stackrel{\partial}{\longrightarrow} \cdots$$

$$\bar{\partial}_E(v)=\mathrm{d} \bar{z}^i \otimes \frac{\partial v^j}{\partial \bar{z}^i} e_j,$$

$$\partial_E(a \cdot v)=\bar{\partial} a \otimes v+a \cdot \partial_E(v) .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|黎曼几何代写Riemannian geometry代考|Higher Order Differential Operators

In the last section, we considered the action of a single vector field on an algebra, and later we will also have actions on modules with connection. However, to have an action we should have an algebra acting, and this requires actions of multiple vector fields giving higher order differential operators. It is reasonable to wish to view these operators as actions of tensor products of vector fields, but such an identification is not entirely trivial. In particular, we will require a connection (called $\bigcirc$ here) on vector fields and 1-forms to achieve this. As we will require several copies of 1-forms and vector fields on $A$, we define
\begin{aligned} &\mathfrak{X}^{\otimes 0}=A, \quad \mathfrak{X}^{\otimes n}=\mathfrak{X} \otimes_A \mathfrak{X} \otimes_A \cdots \otimes_A \mathfrak{X}, \ &\Omega^{1 \otimes 0}=A, \quad \Omega^{1 \otimes n}=\Omega^1 \otimes_A \Omega^1 \otimes_A \cdots \otimes_A \Omega^1, \end{aligned}

where we have $n$ copies of $\mathfrak{X}$ and $\Omega^1$. Note that the definition of $\mathfrak{X}^{\otimes n}$ and $\Omega^{1 \otimes n}$ uses $\otimes_A$, the tensor product over the algebra. We will sometimes use id ${ }^{\otimes n}$ as the identity on $\mathfrak{X}^{\otimes n}$ or $\Omega^{1 \otimes n}$. As for the duality of braided tensor algebras in $\S 2.6$, we define $n$-fold evaluation $\mathrm{ev}^{(n)}: \mathfrak{X}^{\otimes n} \otimes_A \Omega^{1 \otimes n} \rightarrow A$ and $n$-fold coevaluation $\operatorname{coev}^{(n)}: A \rightarrow \Omega^{1 \otimes n} \otimes_A \mathfrak{X}^{\otimes n}$ in a nested way, which we specify recursively by
$$\begin{gathered} \mathrm{ev}^{\langle 1\rangle}=\mathrm{ev}, \quad \mathrm{ev}^{\langle n+1\rangle}=\mathrm{ev}\left(\mathrm{id} \otimes \mathrm{ev}^{(n)} \otimes \mathrm{id}\right) \ \mathrm{coev}^{(1)}=\operatorname{coev}, \quad \operatorname{coev}^{(n+1)}=\left(\text { id } \otimes \operatorname{coev}^{(n)} \otimes \text { id }\right) \operatorname{coev} \end{gathered}$$
In diagrammatic terms, the first of these is
and there is a similar upside down version of this for coevaluation.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Sheaf of Differential Operators DA

So far we have no relations for differential operators corresponding classically to commutativity of partial derivatives. When $T \mathfrak{X}{\bullet}$ is represented on itself, the commutator of covariant derivatives gives curvature, so our first task is to write this as a differential operator, which we do with the help of the torsion $T{\odot}=\mathrm{d}+\wedge \bigcirc$ : $\Omega^1 \rightarrow \Omega^2$, a right module map.

Proposition 6.21 Let $\left(A, \Omega^1\right)$ be an algebra with fgp calculus. There is a central element $\mathcal{R} \in \Omega^2 \otimes_A$ TX๋ given by
$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes f_j \bullet f_i=T_{\odot}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)$$
where $\operatorname{coev}=e_i \otimes f_i \in \Omega^1 \otimes_A \mathfrak{X}$ are dual bases and $e_j \otimes f_j$ is another, such that the curvature $R_{\nabla}$ on $T X_{\bullet}$ in Lemma $6.16$ is given by $R_{\nabla}(\underline{v})=\mathcal{R} \bullet \underline{v}$.

Proof The connection $\nabla$ on $T \mathfrak{X} \bullet$ in Lemma $6.16$ is $\nabla(\underline{v})=e_i \otimes\left(f_i \bullet \underline{v}\right)$ with curvature $R_{\nabla}: T \mathfrak{X}{\bullet} \rightarrow \Omega^2 \otimes_A T$X${\bullet}$ given by
$$R_{\nabla}(\underline{v})=\mathrm{d} e_i \otimes\left(f_i \bullet \underline{v}\right)-e_i \wedge \nabla\left(f_i \bullet \underline{v}\right)=\mathrm{d} e_i \otimes\left(f_i \bullet \underline{v}\right)-e_i \wedge e_j \otimes\left(f_j \bullet f_i \bullet \underline{v}\right) .$$
Now, from the formula for $\bullet$,
$$\begin{gathered} e_i \otimes e_j \otimes f_j \bullet f_i=e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes e_j \otimes(\mathrm{ev} \otimes \mathrm{id})\left(f_j \otimes \otimes f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+\left(\mathrm{id}^{\otimes 2} \otimes \mathrm{ev} \otimes \mathrm{id}\right)\left(\mathrm{id} \otimes \mathrm{coev} \otimes \mathrm{id}^{\otimes 2}\right)\left(e_i \otimes \nabla f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes \nabla f_i=e_i \otimes e_j \otimes f_j \otimes f_i-\nabla e_i \otimes f_i, \end{gathered}$$
where we have used the usual equations for the evaluation and coevaluation. Using the torsion $T_{\odot}$, we can rewrite $\mathcal{R}$ as
$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)+\wedge \otimes e_i \otimes f_i=T_{\bigcirc}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right) .$$
Since $R_{\nabla}$ is a left module map, applying $R_{\nabla}$ to $a \in A$ gives
$$\mathcal{R} \bullet a=R_{\nabla}(a)=R_{\nabla}(a \cdot 1)=a \cdot R_{\nabla}(1)=a \cdot \mathcal{R} \bullet 1=a \cdot \mathcal{R} .$$

## 数学代写|黎曼几何代写黎曼几何代考|高阶微分算子

\begin{aligned} &\mathfrak{X}^{\otimes 0}=A, \quad \mathfrak{X}^{\otimes n}=\mathfrak{X} \otimes_A \mathfrak{X} \otimes_A \cdots \otimes_A \mathfrak{X}, \ &\Omega^{1 \otimes 0}=A, \quad \Omega^{1 \otimes n}=\Omega^1 \otimes_A \Omega^1 \otimes_A \cdots \otimes_A \Omega^1, \end{aligned}

，其中我们有$\mathfrak{X}$和$\Omega^1$的$n$副本。注意$\mathfrak{X}^{\otimes n}$和$\Omega^{1 \otimes n}$的定义使用了$\otimes_A$，即代数上的张量积。我们有时会使用id ${ }^{\otimes n}$作为$\mathfrak{X}^{\otimes n}$或$\Omega^{1 \otimes n}$上的标识。对于编织张量代数$\S 2.6$中的对偶性，我们以嵌套的方式定义了$n$ -fold求值$\mathrm{ev}^{(n)}: \mathfrak{X}^{\otimes n} \otimes_A \Omega^{1 \otimes n} \rightarrow A$和$n$ -fold共求值$\operatorname{coev}^{(n)}: A \rightarrow \Omega^{1 \otimes n} \otimes_A \mathfrak{X}^{\otimes n}$，我们递归指定为
$$\begin{gathered} \mathrm{ev}^{\langle 1\rangle}=\mathrm{ev}, \quad \mathrm{ev}^{\langle n+1\rangle}=\mathrm{ev}\left(\mathrm{id} \otimes \mathrm{ev}^{(n)} \otimes \mathrm{id}\right) \ \mathrm{coev}^{(1)}=\operatorname{coev}, \quad \operatorname{coev}^{(n+1)}=\left(\text { id } \otimes \operatorname{coev}^{(n)} \otimes \text { id }\right) \operatorname{coev} \end{gathered}$$

，还有一个类似的倒挂版本用于共求值

## 数学代写|黎曼几何代写黎曼几何代考|The Sheaf of Differential Operators DA

. The Sheaf of Differential Operators DA

$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes f_j \bullet f_i=T_{\odot}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)$$

$$\begin{gathered} e_i \otimes e_j \otimes f_j \bullet f_i=e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes e_j \otimes(\mathrm{ev} \otimes \mathrm{id})\left(f_j \otimes \otimes f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+\left(\mathrm{id}^{\otimes 2} \otimes \mathrm{ev} \otimes \mathrm{id}\right)\left(\mathrm{id} \otimes \mathrm{coev} \otimes \mathrm{id}^{\otimes 2}\right)\left(e_i \otimes \nabla f_i\right) \ =e_i \otimes e_j \otimes f_j \otimes f_i+e_i \otimes \nabla f_i=e_i \otimes e_j \otimes f_j \otimes f_i-\nabla e_i \otimes f_i, \end{gathered}$$

$$\mathcal{R}=\mathrm{d} e_i \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right)+\wedge \otimes e_i \otimes f_i=T_{\bigcirc}\left(e_i\right) \otimes f_i-e_i \wedge e_j \otimes\left(f_j \otimes f_i\right) .$$

$$\mathcal{R} \bullet a=R_{\nabla}(a)=R_{\nabla}(a \cdot 1)=a \cdot R_{\nabla}(1)=a \cdot \mathcal{R} \bullet 1=a \cdot \mathcal{R} .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

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## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

There are two short topics left to discuss in this chapter, one relatively straightforward and one for discussion. We begin with relative cohomology. Given cochain complexes $(F, \mathrm{~d})$ and $(G, \mathrm{~d})$ with a cochain $\operatorname{map} \phi: F^{n} \rightarrow G^{n}$ for all $n$ (i.e., d $\phi=$ $\phi \mathrm{d})$, we form a new complex $E^{n}=F^{n} \oplus G^{n-1}$ with $\mathrm{d}(f, g)=(\mathrm{d} f, \phi(f)-\mathrm{d} g)$. Then
$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$
We call the cohomology of $(E$, d) the relative cohomology $\mathrm{H}(F, G, \phi)$.
Proposition 4.90 Given a cochain map $\phi: F^{n} \rightarrow G^{n}$ for all $n$ we have a long exact relative cohomology sequence
$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}^{}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi^{}}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots,$$
where $\pi_{1}: E^{n} \rightarrow F^{n}$ is $\pi(f, g)=f$ and $i_{2}: G^{n} \rightarrow E^{n+1}$ is $i_{2}(g)=(-1)^{n}(0, g)$.
Proof Standard algebraic manipulation. Looking at the kernel of $\phi^{*}: \mathrm{H}^{n}(F) \rightarrow$ $\mathrm{H}^{n}(G)$ shows that $\mathrm{H}^{n}(F, G, \phi)$ is defined precisely to make this work.

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framings

Vector bundles in classical geometry typically arise as associated to something deeper, a principal bundle. A connection on this then induces covariant derivatives on all associated bundles in a coherent way. This is the situation in Riemannian geometry where a ‘spin connection’ on the frame bundle induces the Levi-Civita connection on tensor fields but also a covariant derivative on the spinor bundle in the case of a spin manifold, leading to the Dirac operator. Similarly in gauge theory, a principal connection induces covariant derivatives on all associated matter fields.
Briefly, a principal $G$-bundle $P$ over a manifold $X$ is defined exactly like a vector bundle with a surjection $\pi: P \rightarrow X$ but each fibre $P_{x}=\pi^{-1}(x)$ now has the structure of a fixed group $G$. This is achieved by starting with a free right action of $G$ on the manifold $P$ such that $X=P / G$. Free here means any non-identity element of $G$ acts without fixed points, which is equivalent to saying that the map
$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$
is an inclusion, where $p^{g}$ denotes the right action of $g \in G$ on $p \in P$. A connection on $P$ is defined concretely as an equivariant complement in $\Omega^{1}(P)$ to the ‘horizontal forms’ (those pulled back from $\Omega^{1}(X)$ ). This is, however, equivalent to $\omega \in \Omega^{1}(P) \otimes \mathfrak{g}$ with certain properties, where $\mathfrak{g}$ is the Lie algebra of $G$. We will see details in the noncommutative case. Given this data, there is an associated vector bundle $E=P \times_{G} V$ and a connection $\nabla$ on it, for any representation $V$ of $G$. We will give the algebraic and potentially ‘quantum’ version of this notion where the structure group is now a Hopf algebra or ‘quantum group’ as in Chap. 2. We will then use this theory to understand the geometry of quantum homogeneous spaces and framed quantum manifolds more generally.

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$

$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framing

$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

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## 数学代写|黎曼几何代写Riemannian Geometry代考|The Leray–Serre Spectral Sequence

Theorem 4.66 (The Leray-Serre Spectral Sequence) Suppose we are given
(1) a map $\iota: B \longrightarrow A$ which is a differential fibration (see Definition 4.61);
(2) a flat left A-module $E$, with a flat left connection $\nabla_{E}: E \rightarrow \Omega_{A}^{1} \otimes_{A} E$;
(3) the exterior algebra $\Omega_{B}$ has each $\Omega_{B}^{p}$ flat as a right $B$ module.
There is a spectral sequence converging to $\mathrm{H}\left(A, E, \nabla_{E}\right)$ with second page position $(p, q)$ being $\mathrm{H}^{p}\left(B, \hat{\mathrm{H}}^{q}(M), \nabla_{q}\right)$ where $\hat{\mathrm{H}}^{q}(M)$ is the cohomology of
$$\cdots \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q} \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q+1} \stackrel{\mathrm{d}}{\longrightarrow} \cdots$$

with
$$M_{0, q}=\frac{\Omega_{A}^{q} \otimes_{A} E}{\iota \Omega_{B}^{1} \wedge \Omega^{q-1} A \otimes_{A} E}, \quad \mathrm{~d}[x \otimes e]{0, q}=\left[\mathrm{d} x \otimes e+(-1)^{q} x \wedge \nabla{E} e\right]{0, q+1}$$ and $\nabla{q}: \hat{\mathrm{H}}^{q}(M) \rightarrow \Omega_{B}^{1} \otimes_{B} \hat{\mathrm{H}}^{q}(M)$ defined in Lemma 4.65.
Proof The first part of the proof is given in Lemma 4.64. Now we need to calculate the cohomology of
$$\mathrm{d}: \Omega_{B}^{p} \otimes_{B} \hat{\mathrm{H}}^{q}(M) \longrightarrow \Omega_{B}^{p+1} \otimes_{B} \hat{\mathrm{H}}^{q}(M) .$$
If $\xi \in \Omega_{B}^{p}, \eta \in \Omega_{A}^{q}$ and $e \in E$ then $\xi \otimes\langle\eta \otimes e\rangle_{0, q}$ corresponds to $\iota \xi \wedge \eta \otimes e$ and applying $\mathrm{d}$ to the latter gives
$$\iota \mathrm{d} \xi \wedge \eta \otimes e+(-1)^{p} \iota \xi \wedge \mathrm{d} \eta \otimes e+(-1)^{p+q} \iota \xi \wedge \eta \wedge \nabla_{E} e .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Correspondences, Bimodules and Positive Maps

In classical geometry a vector field is both a section of the tangent bundle and a bundle map from the cotangent bundle to the trivial bundle. Geometry is full of things which can be considered both as objects and as maps, and sometimes this can be used to generalise the idea of a mapping. Such is the case for the idea of a correspondence between spaces in topology and algebraic geometry. Roughly speaking (omitting much detail and generalisation), a correspondence between $X$ and $Y$ is a subset $\mathcal{C} \subseteq X \times Y$. In terms of our algebraic picture, the projection to the first coordinate $\pi_{1}: \mathcal{C} \rightarrow X$ gives a map of functions $\pi_{1}^{*}: C(X) \rightarrow C(\mathcal{C})$ and by using this we can regard functions on $\mathcal{C}$ as a module over $C(X)$. Projection to the second coordinate $\pi_{2}: \mathcal{C} \rightarrow Y$ allows us to similarly regard functions on $\mathcal{C}$ as a module over $C(Y)$. These actions commute and $C(\mathcal{C})$ becomes a $C(X)-C(Y)$ bimodule. Tensoring with $C(\mathcal{C})$ gives a functor from $C(Y)$-modules to $C(X)$ modules. This point of view includes the idea of viewing a function $f: X \rightarrow Y$ as a graph ${(x, f(x)) \in X \times Y: x \in X}$, in which case the functor is the pull back. In the noncommutative case we can still consider a $B-A$ bimodule $M$ as a kind of generalised morphism between algebras $A, B$. If we have an actual algebra map $\varphi: A \rightarrow B$ then we construct a $B-A$ bimodule $B_{\varphi}$ by $B_{\varphi}=B$ as a left $B$-module, and right $A$-action given by $b . a=b \varphi(a)$. We have already used this for twisted homology in $\S 3.3 .5$ (albeit the twist in ${ }_{\varsigma} A$ was on the other side) and for the inverse image sheaf in Proposition 4.46. Thus bimodules can be constructed from algebra maps. But we are not limited to this case and can think of a general bimodule in the same spirit as a functor between the algebra representation categories. Bimodules can also be given differentiability properties, as we will see.

Another motivation comes from quantum mechanics, where a measurement on a system gives a projection to an eigenspace of the measurement operator and can be expressed as a completely positive map. We shall focus on the KSGNS construction, which deals with completely positive maps and links them to bimodules.

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|The Leray-Serre Spectral Sequence

(1) 一个映射 $\iota: B \longrightarrow A$ 这是一种微分纤维化（见定义 4.61) ;
(2)一个扁平的左A模块 $E$, 有一个扁平的左连接 $\nabla_{E}: E \rightarrow \Omega_{A}^{1} \otimes_{A} E$;
(3) 外代数 $\Omega_{B}$ 有每个 $\Omega_{B}^{p}$ 平权 $B$ 模块。

$$\cdots \stackrel{\mathrm{d}}{\rightarrow} M_{0, q} \stackrel{\mathrm{d}}{\longrightarrow} M_{0, q+1} \stackrel{\mathrm{d}}{\longrightarrow} \cdots$$

$$M_{0, q}=\frac{\Omega_{A}^{q} \otimes_{A} E}{\iota \Omega_{B}^{1} \wedge \Omega^{q-1} A \otimes_{A} E}, \quad \mathrm{~d}[x \otimes e] 0, q=\left[\mathrm{d} x \otimes e+(-1)^{q} x \wedge \nabla E e\right] 0, q+1$$

$$\mathrm{d}: \Omega_{B}^{p} \otimes_{B} \hat{\mathrm{H}}^{q}(M) \longrightarrow \Omega_{B}^{p+1} \otimes_{B} \hat{\mathrm{H}}^{q}(M) .$$

$$\iota \mathrm{d} \xi \wedge \eta \otimes e+(-1)^{p} \iota \xi \wedge \mathrm{d} \eta \otimes e+(-1)^{p+q} \iota \xi \wedge \eta \wedge \nabla_{E} e .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Correspondences, Bimodules and Positive Maps

KSGNS 构建，它处理完全正映射并将它们链接到双模块。

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

## avatest™帮您通过考试

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## 数学代写|黎曼几何代写Riemannian Geometry代考|Spectral Sequences and Fibrations

A spectral sequence is a machine for doing algebraic calculations in a pictorial manner. This is a large topic and we will give only the briefest of introductions, referring the reader to the excellent text of McCleary for more background. We also restrict our attention to vector spaces and zero entries outside the first quadrant. We use the shorter form im for the image of a map.

Definition 4.50 A double complex of bidegree $(r, t)$ is a collection of vector spaces $E^{n, m}$ (assumed to be zero if either $n<0$ or $m<0$ ) and linear maps

$D: E^{n, m} \rightarrow E^{n+r, m+t}$ with $D \circ D=0$. A spectral sequence of degree $\geq s$ is a collection $\left(E_{r}^{*, *}, D_{r}\right)$ of double complexes of bidegree $(r, 1-r)$ for $r \geq s$ such that
$$E_{r+1}^{n, m} \cong \mathrm{H}^{n, m}\left(E_{r}, D_{r}\right)=\frac{\operatorname{ker} D_{r}: E_{r}^{n, m} \rightarrow E_{r}^{n+r, m+1-r}}{\operatorname{im} D_{r}: E_{r}^{n-r, m+r-1} \rightarrow E_{r}^{n, m}} .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|The Spectral Sequence of a Resolution

A resolution of an object $F$ in an abelian category as in $\S 3.6 .2$ is a collection of objects $E^{n}$ and morphisms forming an exact sequence
$$0 \longrightarrow F \stackrel{\epsilon}{\longrightarrow} E^{0} \stackrel{i_{0}}{\longrightarrow} E^{1} \stackrel{i_{1}}{\longrightarrow} E^{2} \stackrel{i_{2}}{\longrightarrow} \ldots$$
This is sometimes called a right resolution, as opposed to a left resolution with the arrows reversed. Typically the objects $E^{n}$ or maps $i_{n}$ are chosen to have some property which makes calculating with them easier than with the original object $F$. (We have already seen an example, the bar resolution in Definition 3.48.) In our case, we take $F, E^{n} \in{ }{A} \mathcal{F}$ and morphisms $\epsilon, i{0}, \cdots$ and tensor the exact sequence above with $\Omega^{n}$, adding vertical maps, to obtain a double complex

We then take the double complex $C^{n, m}=\Omega^{m} \otimes_{A} E^{n}$ for $n, m \geq 0$ in Example $4.51$, i.e., we throw away the column containing $F$. Note that we have added some minus signs to the connections to satisfy the equation $\mathrm{d}^{\prime \prime} \circ \mathrm{d}^{\prime}+\mathrm{d}^{\prime} \circ \mathrm{d}^{\prime \prime}=0$. Next, keeping the notation of Example $4.51$, if every $\Omega^{n}$ is flat as a right $A$-module then
\begin{aligned} &\mathrm{H}{I}^{n, m}(C)=\mathrm{H}^{n, m}\left(C, d^{\prime}\right)= \begin{cases}\Omega^{m} \otimes{A} F & n=0 \ 0 & n \neq 0,\end{cases} \ &\mathrm{H}{I I}^{n, m}(C)=\mathrm{H}^{n, m}\left(C, d^{\prime \prime}\right)=\mathrm{H}^{n}\left(A, E^{m}, \nabla{E^{m}}\right) . \end{aligned}

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|Spectral Sequences and Fibrations

$D: E^{n, m} \rightarrow E^{n+r, m+t}$ 和 $D \circ D=0$. 度数谱序列 $\geq s$ 是一个集合 $\left(E_{r}^{*, *}, D_{r}\right)$ 双度的双配合物 $(r, 1-r)$ 为了 $r \geq s$ 这样
$$E_{r+1}^{n, m} \cong \mathrm{H}^{n, m}\left(E_{r}, D_{r}\right)=\frac{\operatorname{ker} D_{r}: E_{r}^{n, m} \rightarrow E_{r}^{n+r, m+1-r}}{\operatorname{im} D_{r}: E_{r}^{n-r, m+r-1} \rightarrow E_{r}^{n, m}} .$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|The Spectral Sequence of a Resolution

$$0 \longrightarrow F \stackrel{\epsilon}{\longrightarrow} E^{0} \stackrel{i_{0}}{\longrightarrow} E^{1} \stackrel{i_{1}}{\longrightarrow} E^{2} \stackrel{i_{2}}{\longrightarrow} \ldots$$

$\mathrm{H} I^{n, m}(C)=\mathrm{H}^{n, m}\left(C, d^{\prime}\right)=\left{\Omega^{m} \otimes A F \quad n=00 \quad n \neq 0, \quad \mathrm{H} I I^{n, m}(C)=\mathrm{H}^{n, m}\left(C, d^{\prime \prime}\right)=\mathrm{H}^{n}\left(A, E^{m}, \nabla E^{m}\right) .\right.$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。