## 物理代写|弦论代写string theory代考|MAST90069

statistics-lab™ 为您的留学生涯保驾护航 在代写弦论string theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写弦论string theory代写方面经验极为丰富，各种代写弦论string theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|弦论代写string theory代考|Length, Proper Time, and Reparametrisations

To prepare for the following, we first discuss general curve parameters and reparametrisations. Consider a smooth parametrised curve in Minkowski space,
$$C: I \ni \sigma \longrightarrow x^\mu(\sigma) \in \mathbb{M},$$

where $\sigma$ is an arbitrary curve parameter, taking values in an interval $I \subset \mathbb{R}$ (Figure 1.2).

We can reparametrise the curve by introducing a new curve parameter $\tilde{\sigma} \in \tilde{I}$ which is related to $\sigma$ by an invertible map
$$\sigma \rightarrow \tilde{\sigma}(\sigma) \text {, where } \frac{d \tilde{\sigma}}{d \sigma} \neq 0 .$$
While $C: I \rightarrow \mathbb{M}$ and $\tilde{C}: \tilde{I} \rightarrow \mathbb{M}$ are different maps, they have the same image in $\mathbb{M}$ and we regard them as different descriptions (parametrisations) of the same curve. The quantity $d \tilde{\sigma} / d \sigma$ is the Jacobian of this reparametrisation.
Often, one imposes the stronger condition
$$\frac{d \tilde{\sigma}}{d \sigma}>0,$$
which means that the orientation (direction) of the curve is preserved.
The tangent vector field of the curve is
$$x^{\prime \mu}:=\frac{d x^\mu}{d \sigma} .$$
A curve $C: I \rightarrow \mathbb{M}$ is called space-like, light-like, or space-like if its tangent vector field is space-like, light-like, or space-like, respectively, for all $\sigma \in I$. This property is reparametrisation invariant.

For a space-like curve, $I=\left[\sigma_1, \sigma_2\right] \rightarrow \mathbb{M}$, the length (or ‘proper length’) is defined as
$$L=\int_{\sigma_1}^{\sigma_2} d \sigma \sqrt{\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}} .$$
For a time-like curve, we can define a ‘length’ by
$$\tau\left(\sigma_1, \sigma_2\right)=\int_{\sigma_1}^{\sigma_2} d \sigma \sqrt{-\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}},$$
and this quantity is precisely the proper time for a particle that has this curve as its world-line. We note that the proper length and proper time are distinguished affine curve parameters, characterised by the tangent vector field having unit norm.

## 物理代写|弦论代写string theory代考|A Covariant Action for Massive Relativistic Particles

Using the concepts of the previous section, we introduce the following action:
$$S[x]=-m \int d \sigma \sqrt{-\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}} .$$
Up to the constant factor $-m$, the action is the proper time for the motion of the particle along the world-line. We use an arbitrary curve parameter $\sigma$, and configuration space variables $\left(x, x^{\prime}\right)=\left(x^\mu, x^{\prime \mu}\right)$, which transform covariantly under Lorentz transformations. The action (1.37) has the following symmetries (invariances):

• The action is invariant under reparametrisations $\sigma \rightarrow \tilde{\sigma}(\sigma)$ of the world-line.
• The action is invariant under Poincaré transformations of space-time.
To verify that the new action (1.37) leads to the same field equations as (1.20), we perform the variation $x^\mu \rightarrow x^\mu+\delta x^\mu$ and obtain:
$$\frac{\delta S}{\delta x^\mu}=0 \Leftrightarrow \frac{d}{d \sigma}\left(\frac{m x^{\prime \mu}}{\sqrt{-x^{\prime} \cdot x^{\prime}}}\right)=0 .$$
To get the physical interpretation, we choose the curve parameter $\sigma$ to be the proper time $\tau$ :
$$\frac{d}{d \tau}\left(m \frac{d x^\mu}{d \tau}\right)=m \ddot{x}^\mu=0,$$
where a ‘dot’ denotes the derivative with respect to proper time. This is indeed (1.18) with $f^\mu=0$.

The general solution of this equation, which describes the motion of a free massive particle in Minkowski space is the straight world-line
$$x^\mu(\tau)=x^\mu(0)+\dot{x}^\mu(0) \tau .$$
Remark: Reparametrisations vs Diffeomorphisms. Reparametrisation invariance is also referred to as diffeomorphism invariance. We use the term reparametrisation, rather than diffeomorphism, to emphasise that we interpret the map $\sigma \mapsto \tilde{\sigma}$ passively, that is, as a change parametrisation. In contrast, an active transformation maps a given point to another point. The expressions for passive and active transformation agree up to an overall minus sign, as we will see in later examples, in particular, in Exercise 5.2.2.

## 物理代写|弦论代写string theory代考|Length, Proper Time, and Reparametrisations

$$C: I \ni \sigma \longrightarrow x^\mu(\sigma) \in \mathbb{M},$$

$$\sigma \rightarrow \tilde{\sigma}(\sigma) \text {, where } \frac{d \tilde{\sigma}}{d \sigma} \neq 0 .$$

$$\frac{d \tilde{\sigma}}{d \sigma}>0,$$

$$x^{\prime \mu}:=\frac{d x^\mu}{d \sigma} .$$

$$L=\int_{\sigma_1}^{\sigma_2} d \sigma \sqrt{\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}} .$$

$$\tau\left(\sigma_1, \sigma_2\right)=\int_{\sigma_1}^{\sigma_2} d \sigma \sqrt{-\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}}$$

## 物理代写|弦论代写string theory代考|A Covariant Action for Massive Relativistic Particles

$$S[x]=-m \int d \sigma \sqrt{-\eta_{\mu \nu} \frac{d x^\mu}{d \sigma} \frac{d x^v}{d \sigma}} .$$

• 动作在重新参数化下是不变的 $\sigma \rightarrow \tilde{\sigma}(\sigma)$ 世界线的。
• 该作用在时空庞加莱变换下是不变的。
为了验掯新动作 $(1.37)$ 导致与 $(1.20)$ 相同的场方程，我们执行变分 $x^\mu \rightarrow x^\mu+\delta x^\mu$ 并获得:
$$\frac{\delta S}{\delta x^\mu}=0 \Leftrightarrow \frac{d}{d \sigma}\left(\frac{m x^{\prime \mu}}{\sqrt{-x^{\prime} \cdot x^{\prime}}}\right)=0 .$$
为了得到物理解释，我们选择曲线参数 $\sigma$ 成为适当的时间 $\tau$ :
$$\frac{d}{d \tau}\left(m \frac{d x^\mu}{d \tau}\right)=m \ddot{x}^\mu=0,$$
其中“点“表示关于适当时间的导数。这确实是 (1.18) $f^\mu=0$.
描述自由大质量粒子在 Minkowski 空间中的运动的方程的通解是直线世界线
$$x^\mu(\tau)=x^\mu(0)+\dot{x}^\mu(0) \tau .$$
备注: 重新参数化与微分同胚。重新参数化不变性也称为微分同胚不变性。我们使用术语重新参数化而不是微分同 胚来强调我们解释地图 $\sigma \mapsto \tilde{\sigma}$ 被动地，即作为更改参数化。相反，主动变换将给定点映射到另一个点。正如我们 将在后面的例子中看到的，特别是在练习 $5.2 .2$ 中，被动和主动变换的表达式一致为一个负号。

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## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|弦论代写string theory代考|PHY-897S

statistics-lab™ 为您的留学生涯保驾护航 在代写弦论string theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写弦论string theory代写方面经验极为丰富，各种代写弦论string theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|弦论代写string theory代考|A Non-covariant Action Principle for Relativistic Particles

The equations of motion of all fundamental physical theories can be obtained from variational principles. In this approach, a theory is defined by specifying its action which is a functional on the configuration space. The equations of motion are the Euler-Lagrange equations obtained by imposing that the action is invariant under infinitesimal variations of the path, with the initial and final position kept fixed.

For a point particle, the configuration space is parametrised by its position $\vec{x}$ and velocity $\vec{v}$. The action functional takes the form
$$S[\vec{x}]=\int d t L(\vec{x}(t), \vec{v}(t))$$
In principle, the Lagrangian $L$ can have an explicit dependence on time, corresponding to a time-dependent potential or external field. In fundamental theories, we assume the invariance of the field equations under time-translations, which forbids an explicit time dependence of $L$.

The action for a free, massive, relativistic particle is proportional to the proper time along the world-line, and given by minus the product of its mass and the proper time:
$$S=-m \int d t \sqrt{1-\vec{v}^2} .$$
The minus sign has been introduced so that $L$ has the conventional form $L=T-V$ where $T$ is the part quadratic in time derivatives, that is, the kinetic energy. The remaining part $V$ is the potential energy. We work in units where the speed of light and the reduced Planck constant have been set to unity, $c=1, \hbar=1$. In such natural units the action $S$ is dimensionless. To verify that the action principle reproduces the equation of motion (1.18), we consider the motion $\vec{x}(t)$ of a particle between the initial postion $\vec{x}_1=\vec{x}\left(t_1\right)$ and the final position $\vec{x}_2=\vec{x}\left(t_2\right)$.

## 物理代写|弦论代写string theory代考|Canonical Momenta and Hamiltonian

We now turn to the Hamiltonian description of the relativistic particle. In the Lagrangian formalism we use the configuration space variables $(\vec{x}, \vec{v})=\left(x^i, v^i\right)$. In the Hamiltonian formalism, the velocity $\vec{v}$ is replaced by the canonical momentum
$$\pi^i:=\frac{\partial L}{\partial v_i} .$$
For the Lagrangian $L=-m \sqrt{1-\vec{v}^2}$, the canonical momentum agrees with the kinetic momentum, $\vec{\pi}=\vec{p}=\left(1-\vec{v}^2\right)^{-1 / 2} m \vec{v}$. However, conceptually canoncial and kinetic momentum are different quantities. A standard example where the two quantities are not equal is a charged particle in a magnetic field (see Section 13.6, i.p. formula (13.169)).

The Hamiltonian $H(\vec{x}, \vec{\pi})$ is obtained from the Lagrangian $L(\vec{x}, \vec{v})$ by a Legendre transformation:
$$H(\dot{x}, \vec{\pi})=\vec{\pi} \cdot \dot{v}-L(\dot{x}, \dot{v}(\dot{x}, \vec{\pi})) .$$
For $L=-m \sqrt{1-\vec{v}^2}$ the Hamiltonian is equal to the total energy:
$$H=\vec{\pi} \cdot \vec{v}-L=\vec{p} \cdot \vec{v}-L=\frac{m}{\sqrt{1-\vec{v}^2}}=p^0=E .$$
Describing relativistic particles using the action (1.20) has the following disadvantages:

• We can describe massive particles, but photons, gluons, and the hypothetical gravitons underlying gravity are believed to be massless. How can we describe massless particles?
• The independent variables $\vec{x}, \vec{v}$ are not Lorentz vectors. Therefore, our formalism lacks manifest Lorentz covariance. How can we formulate an action principle that is Lorentz covariant?
• We have picked a particular curve parameter for the world-line, namely the inertial time with respect to a Lorentz frame. While this is a natural choice, ‘physics’, that is, observational data, cannot depend on how we label points on the world-line. How can we formulate an action principle that is manifestly covariant with respect to reparametrisations of the world-line?

## 物理代写|弦论代写string theory代考|A Non-covariant Action Principle for Relativistic Particles

$$S[\vec{x}]=\int d t L(\vec{x}(t), \vec{v}(t))$$

$$S=-m \int d t \sqrt{1-\vec{v}^2} .$$

## 物理代写|弦论代写string theory代考|Canonical Momenta and Hamiltonian

$$\pi^i:=\frac{\partial L}{\partial v_i} .$$

$$H(\dot{x}, \vec{\pi})=\vec{\pi} \cdot \dot{v}-L(\dot{x}, \dot{v}(\dot{x}, \vec{\pi})) .$$

$$H=\vec{\pi} \cdot \vec{v}-L=\vec{p} \cdot \vec{v}-L=\frac{m}{\sqrt{1-\vec{v}^2}}=p^0=E .$$

• 我们可以描述大质量的粒子，但光子、胶子和假设的引力背后的引力子被认为是无质量的。我们如何描述无 质量粒子?
• 自变量 $\vec{x}, \vec{v}$ 不是洛伦兹向量。因此，我们的形式主义缺乏明显的洛伦兹协方差。我们如何制定一个洛伦兹协 变的作用原理?
• 我们为世界线选择了一个特定的曲线参数，即相对于洛伦兹框架的惯性时间。虽然这是一个自然的选择，但 “物理学”，即观测数据，不能依赖于我们如何在世界线上标记点。我们如何制定一个与世界线重新参数化相 关的明显协变的行动原则?

﻿

﻿

﻿

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

﻿

## 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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|弦论代写string theory代考|PHY622

statistics-lab™ 为您的留学生涯保驾护航 在代写弦论string theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写弦论string theory代写方面经验极为丰富，各种代写弦论string theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|弦论代写string theory代考|Minkowski Space

According to Einstein’s theory of special relativity, space and time are combined into ‘space-time’, which is modelled by Minkowski space $M$. ‘ The elements $P, Q, \ldots \in$ $\mathbb{M}$ are called events. We leave the dimension $D$ of space-time unspecified. Minkowski space is homogeneous and thus has no preferred origin, which makes it a point space (affine space) rather than a vector space (linear space). However, displacements relating events $P, Q$ are vectors,
$$x=\overrightarrow{P Q} \in \mathbb{R}^D,$$
and once we choose a point $O \in \mathbb{M}$ as the origin of our coordinate system there is a one-to-one correspondence between events $P$ and position vectors
$$x_P=\overrightarrow{O P} \text {. }$$
The components
$$\left(x^\mu\right){\mu=0,1, \ldots, D-1}=\left(x^0, \vec{x}\right), \quad \vec{x}=\left(x^i\right){i=1, \ldots, D-1}$$
of vectors $x \in \mathbb{R}^D$ provide linear coordinates on $\mathbb{M}$. We assume that $x^i=0$ is the world-line of an inertial (force-free) observer, so that $x^0=c t$ is proportional to the time $t$ measured in the associated inertial system, while $x^i$ provide linear coordinates on space. We will normally use natural units where we set the speed of light to unity, $c=1 .^2$
To measure the distance between events, we use the indefinite scalar product
$$x \cdot y=\eta_{\mu v} x^\mu y^v,$$
on the vector space $\mathbb{R}^D$, with Gram matrix
$$\eta=\left(\eta_{\mu \nu}\right)=\left(\begin{array}{cc} -1 & \overrightarrow{0}^T \ \overrightarrow{0} & \mathbb{1}_{D-1} \end{array}\right) .$$

## 物理代写|弦论代写string theory代考|Particles

The fundamental constituents of matter are usually modelled as particles, that is, as objects that are localised and can be characterised by a small number of parameters, such as mass, spin, and charges. While some particles are bound states of others, the standard model of particle physics is based on a list of particles, assumed to be elementary in the sense that they do not have constituents and, therefore, no internal excitations. In classical mechanics, such particles are modelled as mathematical points. The motion of such a point particle, or particle for short, is described by a parametrised curve called the world-line. If we restrict ourselves to inertial frames, it is natural to choose the coordinate time $t$ as the curve parameter. Then, the world-line of a particle is a parametrised curve
$$C: I \rightarrow \mathbb{M}: t \mapsto x(t)=\left(x^\mu(t)\right)=(t, \vec{x}(t)),$$
where $I \subset \mathbb{R}$ is the time interval for which the particle is observed. $I=\mathbb{R}$ is included as a limiting case.
The velocity of a particle relative to an inertial frame is
$$\vec{v}=\frac{d \vec{x}}{d t},$$
and $v=\sqrt{\vec{v} \cdot \vec{v}} \geq 0$ is the speed. Since $t$ and $\vec{v}$ are not covariant quantities (Lorentz tensors), it is useful to formulate relativistic mechanics using the Lorentz vector $x^\mu$ and its derivatives with respect to a curve parameter which is a Lorentz scalar. This works differently for massive and for massless particles.

The inertial mass $m$ of a particle measures its resistance against a change of velocity. Massive particles, $m>0$, propagate with velocities $v<1$ and have timelike world-lines, that is world-lines where the tangent vector is time-like everywhere. Massless particles, $m=0$, propagate with velocity $v=1$ and have light-like world-lines. Poincaré symmetry also admits tachyons, that is, particles with negative mass-squared, $m^2<0$, which propagate with velocity $v>1$ and have space-like world-lines. Such tachyons are discarded because they would allow a-causal effects, such as sending signals backwards in time. In quantum field theory, tachyons are re-interpreted as indicating instabilities resulting from expanding a theory around a local maximum of the potential. This is a physical effect and does not involve particles propagating with superluminal speed (see Section 7.7).

## 物理代写|弦论代写string theory代考|Minkowski Space

$$x=\overrightarrow{P Q} \in \mathbb{R}^D,$$

$$x_P=\overrightarrow{O P} \text {. }$$

$$\left(x^\mu\right) \mu=0,1, \ldots, D-1=\left(x^0, \vec{x}\right), \quad \vec{x}=\left(x^i\right) i=1, \ldots, D-1$$

$$x \cdot y=\eta_{\mu v} x^\mu y^v,$$

## 物理代写|弦论代写string theory代考|Particles

$$C: I \rightarrow \mathbb{M}: t \mapsto x(t)=\left(x^\mu(t)\right)=(t, \vec{x}(t)),$$

$$\vec{v}=\frac{d \vec{x}}{d t},$$

﻿

﻿

﻿

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

﻿

## 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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

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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。