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# 数学代写|谱几何代写Spectral Geometry代考|MATH741 Inverse problem: Can you hear the shape of a drum?

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## 数学代写|谱几何代写Spectral Geometry代考|Inverse problem: Can you hear the shape of a drum?

Inverse problem: If I know (more or less) the Laplace eigenvalues of a domain, what can I deduce of its geometry?

Suppose you have perfect pitch. Could you derive the shape of a drum from the music you hear from it? More generally, can you determine the structural soundness of an object by listening to its vibrations? This question was first posed by Schuster in 1882. As Berger says in his book A panoramic view of Riemannian Geometry,
“Already in the middle ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. How can one test the resistance to vibrations of large modern structures by nondestructive essays?… A small crack will not only change the boundary shape of our domain, one side of the crack will strike the other during vibrations invalidating our use of the simple linear wave equation. On the other hand, heat will presumably not leak out of a thin crack very quickly, so perhaps the heat equation will still provide a reasonable approximation for a short time…”
An infinite sequence of numbers determines via Fourier analysis an integrable function. It wouldn’t be that crazy if an infinite sequences of eigenvalues would determine the shape of the domain. Unfortunately, the answer to the question can you hear the shape of a drum? is no. This was proved in 1992 by Gordon, Web and Wolpert. Nowadays many planar domains are known to have different shapes but exactly the same spectrum.

## 数学代写|谱几何代写Spectral Geometry代考|Hearing the length of a guitar string

Consider an interval $[0, \ell]$ with Dirichlet boundary conditions $\varphi(0)=\varphi(\ell)=0$.
The eigenfunctions are
$$\varphi_k(x)=\sin \left(\frac{k \pi}{\ell} x\right) \quad \text { for } k \geq 1$$
with eigenvalues $\lambda_k=\left(\frac{k \pi}{\ell}\right)^2$ for $k \geq 1$.

After the first half of the 18th century mathematicians such as d’Alembert and Bernoulli developed the theory of a vibrational string. As one should expect, the vibrations of a string will depend on many factors such us its length, mass and tension.

To simplify our exposition consider a guitar string of length $\ell$ which we model as the interval $[0, \ell]$. Assume further that the density mass and the tension are constant and equal to 1 . Today it comes as no surprise that the behavior of a vibrating string is described by the wave equation. That is, if we write $x$ for a point in the string $[0, \ell]$ and $t$ for the time variable, then the height $u(x, t)$ of the string above the point $x$ after a time $t$ should satisfy the wave equation
$$\Delta u(x, t)=\frac{\partial^2}{\partial t^2} u(x, t) .$$

## 数学代写|谱几何代写Spectral Geometry代考|Inverse problem: Can you hear the shape of a drum?

“早在中世纪，制钟人就知道如何通过在地面上敲响大钟，然后再把它抬到钟楼上，来检测看不见的裂缝。如何通过无损检测论文来测试大型现代结构的抗震性呢……一个小裂缝不仅会改变我们领域的边界形状，裂缝的一边会在振动时撞击另一边，使我们对简单线性波方程的使用失效。另一方面，热量大概不会很快从细小的裂缝中泄漏出来，所以也许热方程在短时间内仍会提供一个合理的近似值……”

## 数学代写|谱几何代写Spectral Geometry代考|Hearing the length of a guitar string

$$\varphi_k(x)=\sin \left(\frac{k \pi}{\ell} x\right) \quad \text { for } k \geq 1$$

18 世纪上半叶之后，达朗贝尔和伯努利等数学家发展了弦振动理论。正如人们应该预料的那样，一根弦的振动将取诀于许多因 㨞，例如的长度、质量和张力。

$$\Delta u(x, t)=\frac{\partial^2}{\partial t^2} u(x, t) .$$

## 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。