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## 数学代写|数学物理代写Mathematical Physics代考|b-Fibration Algebra

Returning to the initial setting of a fibration the opposite extreme to the semiclassical case corresponds to a trivial fibration in the sense that $\phi$ has one fibre. Then the operators are simply smooth in the parameter $t$. This serves to emphasize that the map to $[0,1]$ is itself a fibration and this can be generalized – as in the setting of the Atiyah-Singer index theorem – to the case of a more general base and fibration with $X \times[0,1]$ replace by a fibration $\phi: \hat{X} \longrightarrow B$ with typical fibre $X$. This in turn can be generalized to the case of a $b$-fibration which allows degeneration, of a specific type, on the fibres. Rather than define this in general let me point to a specific type of example.

Suppose $M$ is a compact manifold with corners. It may have many boundary hypersurfaces but each has (by assumption) a defining function – a smooth nonnegative function on $M$ which vanishes precisely at the boundary hypersurface in question and has non-zero differential there. Then a total boundary defining function on $M$ is a product of such functions. More generally one can take a product of positive integral powers of such functions. The resulting function, which vanishes at every boundary point but is positive elsewhere, defines a b-fibration – a kind of collar decomposition – as a map to $[0, \epsilon]$ for $\epsilon>0$ small enough. The general case of a b-fibration is locally the product of such maps and an ordinary fibration. In any case there is a similar structure to the case of a fibration if vector fields tangent to all boundaries are considered and there is an algebra of pseudodifferential operators reducing to the fibre pseudodifferential operators for a fibration:
$$\mathcal{V}{\mathrm{b} / \phi}(M) \subset \rho^\alpha \operatorname{Diff}{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^{* *}(M ; W)$$
The last space here actually depends on a choice of the resolution of the fibre diagonal ([3]).

Example of this is provided by the blow-up calculus being developed with Pierre Albin and also the gluing calculus with Michael Singer which corresponds to gluing problems such as treated by Arezzo and Pacard [1]. Namely, from a manifold with an interior separating hypersurface a manifold with corners can be constructed with b-fibration which corresponds to the process of gluing a complete metric on one side of the hypersurface to an appropriate (often incomplete) metric on the other side. The b-fibration is of the type discussed above. This is also related to older work with Andrew Hassell and Rafe Mazzeo on the eta invariant.

## 数学代写|数学物理代写Mathematical Physics代考|Morse Degeneration

The two types of calculus above, corresponding to a b-fibration, where the vector fields degenerate only at a submanifold in the boundary, and the adiabatic case where the degenerate to be tangent to the fibres of a fibration can be combined. Rather than set this up in general – corresponding to iterated b-fibrations where there are finer fibrations over (some of) the boundary hypersurfaces of the first b-fibration – let me simply indicate an example which arises from a question of Atiyah.

Every compact manifold $M$ carries a Morse function $f: M \longrightarrow[a, b]$. This can be thought of as a generalization of a fibration over the circle – the setting considered by Witten in [4]. There are singular fibres but they are isolated and of ‘minimal singularity’. In particular the singular points, where the differential of $f$ vanishes, are themselves isolated. To construct an adiabatic limit of this b-type, first replace $M$ by the manifold with boundary in which the critical points of $f$ are blown up radially, $M_{\mathrm{C}}=[M ; \mathrm{Cp}]$ to which $f$ lifts as a smooth function. The singular fibres of $f$ are resolved in the sense that they are each the union of a boundary hypersurface and an embedded (generally non-connected) submanifold SF which meets this boundary transversally. The full space with b-fibration we consider is
$$\left[M_{\mathrm{C} P} \times[0,1]_t ; \mathrm{SF}\right] \longrightarrow[0,1]_t .$$
The additional condition imposes on vector fields (and hence differential operators) corresponding to the adiabatic limit is that over $t=0$ they should be tangent to the boundaries, to the regular fibres of $f$ and to the fibres of the blow-down map for the blow-up of SF.

Atiyah’s question is whether for a Dirac operator on the total space one can find a formula for the index (which of course is known) in terms of the spectral flow of the induced Dirac operators on the fibres, between the singular values (likely regularized in some way) with perhaps some ‘jump terms’ across the singular fibres. For the moment I only know how do do this after perturbing the operator by a smoothing operator associated to the calculus that I have implicitly described above. To give a more realistic answer requires a better understanding of the behaviour of the eta invariant.

# 数学物理代写

## 数学代写|数学物理代写数学物理代考|b-纤维代数

. b . b

$$\mathcal{V}{\mathrm{b} / \phi}(M) \subset \rho^\alpha \operatorname{Diff}{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^{* *}(M ; W)$$

## 数学代写|数学物理代写数学物理代考|莫尔斯退化

.

$$\left[M_{\mathrm{C} P} \times[0,1]_t ; \mathrm{SF}\right] \longrightarrow[0,1]_t .$$与绝热极限相对应的向量场(因此是微分算子)的附加条件是over $t=0$ 它们应该与边界相切，与规则的纤维相切 $f$ 和到吹倒图的纤维为吹倒的SF.

Atiyah的问题是，对于总空间上的狄拉克算符，人们是否可以找到一个指数公式(这当然是已知的)，根据纤维上的诱导狄拉克算符的光谱流，在奇异值(可能以某种方式正则化)之间，可能有一些跨越奇异纤维的“跳跃项”。到目前为止，我只知道在用一个与微积分相关的平滑算子扰动算子之后如何做到这一点，我已经在上面隐式地描述过了。要给出一个更现实的答案，需要对eta不变量的行为有更好的理解

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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。