Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法

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## 数学代写|数学物理代写Mathematical Physics代考|Degenerate Fibres: Analytic Case

One natural question is what happens to the adiabatic limit if the form of the metric is generalized. This is related to issues below and there are several levels of ‘relaxation’ of the conditions. One might first consider perturbations of first order in $t$ which include cross terms between base and fibre such as $t d y \otimes d z$. Note that it does not really make sense to want these to have coefficients which only depend on the base variables. The effect of adding such a term can be quite dramatic, since it can change the invertibility properties of the adiabatic model operators although this is still an elliptic suspended family – the null spaces may no longer form a bundle, as they may not be smooth over the base. If the failure to be smooth is itself reasonably smooth, as discussed below, then something can be done. I don’t really know what happens to the null space in the general case – if anyone wants to try to work it out they are welcome to try and I am interested to discuss it!

Another possible ‘perturbation’ is to replace $h$ by a basic tensor – that is, to let its coefficients vary on the fibre. This is actually less of a problem than the introduction of cross terms and leads to a very similar structure, but the details have not been written up as far as I know.

Next consider the effect of adding a potential, for simplicity in the case of the Laplacian on functions. The simplest case is when the potential is real and nonnegative. If it is non-zero on any fibre then the model on that fibre is invertible. The opposite extreme to uniform degeneracy is when $V$ vanishes on isolated fibres and has non-zero Hessian (in the base variables) at every point on those fibres

Then $\Delta_t+V$ has an inverse with a weaker (optimal) bound than in a case such as (4) when the model operators are invertible
$$\left|\left(\Delta_t+V\right)^{-1}\right|_{L^2} \leq C t$$
In this case there are additional model operators at the singular fibres which are harmonic oscillators.

## 数学代写|数学物理代写Mathematical Physics代考|Eigenvalues for Triangles

One geometric setting closely related to the perturbation by a potential with isolated minima is the vertical collapse of a manifold with boundary. For example, if one takes a region in the plane between two $2 \pi$-periodic smooth curves, $\Omega=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y<U(x)\right}$, and ‘collapses’ it by vertical scaling to $\Omega_t=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y / t<U(x)\right}, 0<t \leq 1$, then consider the behaviour of the eigenvalues for the Dirichlet, or Neumann, problem. For the Dirichlet problem the smallest eigenvalue has an asymptotic expansion related to that above, in particular there are harmonic oscillator models, provided $U(x)-L(x)$ only has non-degenerated maxima.

The general problem of the behaviour of the eigenvalues for the Dirichlet problem for triangles as functions on moduli space remains open and certainly there is behaviour similar to this under vertical collapse, except that the harmonic oscillator is replaced by its ‘one-sided’ cousion, Airy’s operator.

# 数学物理代写

## 数学代写数学物理代写Mathematical Physics代考|Degenerate Fibres: Analytic Case

$$\left|\left(\Delta_t+V\right)^{-1}\right|_{L^2} \leq C t$$

## 数学代写数学物理代写Mathematical Physics代考|Eigenvalues for Triangles

\eft 的分隔符缺失或无法识别，然后考虑 Dirichlet 或 Neumann 问题的特征值的行为。对于狄利克雷问

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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。