text:
## 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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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。

## 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|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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

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。


text:
# Fight Finance

#### CoursesTagsRandomAllRecentScores

Your friend just bought a house for $400,000. He financed it using a$320,000 mortgage loan and a deposit of $80,000. In the context of residential housing and mortgages, the 'equity' tied up in the value of a person's house is the value of the house less the value of the mortgage. So the initial equity your friend has in his house is$80,000. Let this amount be E, let the value of the mortgage be D and the value of the house be V. So $V=D+E$.

If house prices suddenly fall by 10%, what would be your friend's percentage change in equity (E)? Assume that the value of the mortgage is unchanged and that no income (rent) was received from the house during the short time over which house prices fell.

Remember:

$$r_{0\rightarrow1}=\frac{p_1-p_0+c_1}{p_0}$$

where $r_{0-1}$ is the return (percentage change) of an asset with price $p_0$ initially, $p_1$ one period later, and paying a cash flow of $c_1$ at time $t=1$.

Two risky stocks A and B comprise an equal-weighted portfolio. The correlation between the stocks' returns is 70%.

If the variance of stock A increases but the:

• Prices and expected returns of each stock stays the same,
• Variance of stock B's returns stays the same,
• Correlation of returns between the stocks stays the same.

Which of the following statements is NOT correct?

All things remaining equal, the higher the correlation of returns between two stocks:

One and a half years ago Frank bought a house for $600,000. Now it's worth only$500,000, based on recent similar sales in the area.

The expected total return on Frank's residential property is 7% pa.

He rents his house out for $1,600 per month, paid in advance. Every 12 months he plans to increase the rental payments. The present value of 12 months of rental payments is$18,617.27.

The future value of 12 months of rental payments one year in the future is $19,920.48. What is the expected annual rental yield of the property? Ignore the costs of renting such as maintenance, real estate agent fees and so on. Which of the following statements about cash in the form of notes and coins is NOT correct? Assume that inflation is positive. Notes and coins: How can a nominal cash flow be precisely converted into a real cash flow? You expect a nominal payment of$100 in 5 years. The real discount rate is 10% pa and the inflation rate is 3% pa. Which of the following statements is NOT correct?

An investor bought a bond for $100 (at t=0) and one year later it paid its annual coupon of$1 (at t=1). Just after the coupon was paid, the bond price was $100.50 (at t=1). Inflation over the past year (from t=0 to t=1) was 3% pa, given as an effective annual rate. Which of the following statements is NOT correct? The bond investment produced a: You're considering making an investment in a particular company. They have preference shares, ordinary shares, senior debt and junior debt. Which is the safest investment? Which will give the highest returns? Who is most in danger of being personally bankrupt? Assume that all of their businesses' assets are highly liquid and can therefore be sold immediately. Which of the following statements about book and market equity is NOT correct? A newly floated farming company is financed with senior bonds, junior bonds, cumulative non-voting preferred stock and common stock. The new company has no retained profits and due to floods it was unable to record any revenues this year, leading to a loss. The firm is not bankrupt yet since it still has substantial contributed equity (same as paid-up capital). On which securities must it pay interest or dividend payments in this terrible financial year? What is the lowest and highest expected share price and expected return from owning shares in a company over a finite period of time? Let the current share price be $p_0$, the expected future share price be $p_1$, the expected future dividend be $d_1$ and the expected return be $r$. Define the expected return as: $r=\dfrac{p_1-p_0+d_1}{p_0}$ The answer choices are stated using inequalities. As an example, the first answer choice "(a) $0≤p<∞$ and $0≤r< 1$", states that the share price must be larger than or equal to zero and less than positive infinity, and that the return must be larger than or equal to zero and less than one. Which of the following statements is NOT correct? The Australian Federal Government lends money to domestic students to pay for their university education. This is known as the Higher Education Contribution Scheme (HECS). The nominal interest rate on the HECS loan is set equal to the consumer price index (CPI) inflation rate. The interest is capitalised every year, which means that the interest is added to the principal. The interest and principal does not need to be repaid by students until they finish study and begin working. Which of the following statements about HECS loans is NOT correct? The following is the Dividend Discount Model (DDM) used to price stocks: $$P_0=\dfrac{C_1}{r-g}$$ If the assumptions of the DDM hold, which one of the following statements is NOT correct? The long term expected: A stock will pay you a dividend of$10 tonight if you buy it today. Thereafter the annual dividend is expected to grow by 5% pa, so the next dividend after the $10 one tonight will be$10.50 in one year, then in two years it will be $11.025 and so on. The stock's required return is 10% pa. What is the stock price today and what do you expect the stock price to be tomorrow, approximately? The following equation is called the Dividend Discount Model (DDM), Gordon Growth Model or the perpetuity with growth formula: $$P_0 = \frac{ C_1 }{ r - g }$$ What is $g$? The value $g$ is the long term expected: Most listed Australian companies pay dividends twice per year, the 'interim' and 'final' dividends, which are roughly 6 months apart. You are an equities analyst trying to value the company BHP. You decide to use the Dividend Discount Model (DDM) as a starting point, so you study BHP's dividend history and you find that BHP tends to pay the same interim and final dividend each year, and that both grow by the same rate. You expect BHP will pay a$0.55 interim dividend in six months and a $0.55 final dividend in one year. You expect each to grow by 4% next year and forever, so the interim and final dividends next year will be$0.572 each, and so on in perpetuity.

Assume BHP's cost of equity is 8% pa. All rates are quoted as nominal effective rates. The dividends are nominal cash flows and the inflation rate is 2.5% pa.

What is the current price of a BHP share?

You are an equities analyst trying to value the equity of the Australian telecoms company Telstra, with ticker TLS. In Australia, listed companies like Telstra tend to pay dividends every 6 months. The payment around August is called the final dividend and the payment around February is called the interim dividend. Both occur annually.

• Today is mid-March 2015.
• TLS's last interim dividend of $0.15 was one month ago in mid-February 2015. • TLS's last final dividend of$0.15 was seven months ago in mid-August 2014.

Judging by TLS's dividend history and prospects, you estimate that the nominal dividend growth rate will be 1% pa. Assume that TLS's total nominal cost of equity is 6% pa. The dividends are nominal cash flows and the inflation rate is 2.5% pa. All rates are quoted as nominal effective annual rates. Assume that each month is exactly one twelfth (1/12) of a year, so you can ignore the number of days in each month.

Calculate the current TLS share price.

Two companies BigDiv and ZeroDiv are exactly the same except for their dividend payouts.

BigDiv pays large dividends and ZeroDiv doesn't pay any dividends.

Currently the two firms have the same earnings, assets, number of shares, share price, expected total return and risk.

Assume a perfect world with no taxes, no transaction costs, no asymmetric information and that all assets including business projects are fairly priced and therefore zero-NPV.

All things remaining equal, which of the following statements is NOT correct?

A stock is expected to pay a dividend of $15 in one year (t=1), then$25 for 9 years after that (payments at t=2 ,3,...10), and on the 11th year (t=11) the dividend will be 2% less than at t=10, and will continue to shrink at the same rate every year after that forever. The required return of the stock is 10%. All rates are effective annual rates.

What is the price of the stock now?

Carlos and Edwin are brothers and they both love Holden Commodore cars.

Carlos likes to buy the latest Holden Commodore car for $40,000 every 4 years as soon as the new model is released. As soon as he buys the new car, he sells the old one on the second hand car market for$20,000. Carlos never has to bother with paying for repairs since his cars are brand new.

Edwin also likes Commodores, but prefers to buy 4-year old cars for $20,000 and keep them for 11 years until the end of their life (new ones last for 15 years in total but the 4-year old ones only last for another 11 years). Then he sells the old car for$2,000 and buys another 4-year old second hand car, and so on.

Every time Edwin buys a second hand 4 year old car he immediately has to spend $1,000 on repairs, and then$1,000 every year after that for the next 10 years. So there are 11 payments in total from when the second hand car is bought at t=0 to the last payment at t=10. One year later (t=11) the old car is at the end of its total 15 year life and can be scrapped for $2,000. Assuming that Carlos and Edwin maintain their love of Commodores and keep up their habits of buying new ones and second hand ones respectively, how much larger is Carlos' equivalent annual cost of car ownership compared with Edwin's? The real discount rate is 10% pa. All cash flows are real and are expected to remain constant. Inflation is forecast to be 3% pa. All rates are effective annual. Ignore capital gains tax and tax savings from depreciation since cars are tax-exempt for individuals. You own a nice suit which you wear once per week on nights out. You bought it one year ago for$600. In your experience, suits used once per week last for 6 years. So you expect yours to last for another 5 years.

Your younger brother said that retro is back in style so he wants to wants to borrow your suit once a week when he goes out. With the increased use, your suit will only last for another 4 years rather than 5.

What is the present value of the cost of letting your brother use your current suit for the next 4 years?

Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new suit when your current one wears out and your brother will not use the new one; your brother will only use your current suit so he will only use it for the next four years; and the price of a new suit never changes.

Which of the below statements about effective rates and annualised percentage rates (APR's) is NOT correct?

On his 20th birthday, a man makes a resolution. He will deposit $30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity. The bank account pays interest at 6% pa compounding monthly, which is not expected to change. If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment? You want to buy an apartment priced at$300,000. You have saved a deposit of $30,000. The bank has agreed to lend you the$270,000 as an interest only loan with a term of 25 years. The interest rate is 12% pa and is not expected to change.

What will be your monthly payments? Remember that mortgage payments are paid in arrears (at the end of the month).

You just borrowed $400,000 in the form of a 25 year interest-only mortgage with monthly payments of$3,000 per month. The interest rate is 9% pa which is not expected to change.

You actually plan to pay more than the required interest payment. You plan to pay $3,300 in mortgage payments every month, which your mortgage lender allows. These extra payments will reduce the principal and the minimum interest payment required each month. At the maturity of the mortgage, what will be the principal? That is, after the last (300th) interest payment of$3,300 in 25 years, how much will be owing on the mortgage?

A bank grants a borrower an interest-only residential mortgage loan with a very large 50% deposit and a nominal interest rate of 6% that is not expected to change. Assume that inflation is expected to be a constant 2% pa over the life of the loan. Ignore credit risk.

From the bank's point of view, what is the long term expected nominal capital return of the loan asset?

An 'interest only' loan can also be called a:

The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.

Considering this, which of the following statements is NOT correct?

In these tough economic times, central banks around the world have cut interest rates so low that they are practically zero. In some countries, government bond yields are also very close to zero.

A three year government bond with a face value of $100 and a coupon rate of 2% pa paid semi-annually was just issued at a yield of 0%. What is the price of the bond? An Australian company just issued two bonds: • A 1 year zero coupon bond at a yield of 8% pa, and • A 2 year zero coupon bond at a yield of 10% pa. What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted. You're trying to save enough money to buy your first car which costs$2,500. You can save $100 at the end of each month starting from now. You currently have no money at all. You just opened a bank account with an interest rate of 6% pa payable monthly. How many months will it take to save enough money to buy the car? Assume that the price of the car will stay the same over time. Which of the following statements about gold is NOT correct? Assume that the gold price increases by inflation. Gold: A stock’s current price is$1. Its expected total return is 10% pa and its long term expected capital return is 4% pa. It pays an annual dividend and the next one will be paid in one year. All rates are given as effective annual rates. The dividend discount model is thought to be a suitable model for the stock. Ignore taxes. Which of the following statements about the stock is NOT correct?

A share’s current price is $60. It’s expected to pay a dividend of$1.50 in one year. The growth rate of the dividend is 0.5% pa and the stock’s required total return is 3% pa. The stock’s price can be modeled using the dividend discount model (DDM):

$P_0=\dfrac{C_1}{r-g}$

Which of the following methods is NOT equal to the stock’s expected price in one year and six months (t=1.5 years)? Note that the symbolic formulas shown in each line below do equal the formulas with numbers. The formula is just repeated with symbols and then numbers in case it helps you to identify the incorrect statement more quickly.

A share currently worth $100 is expected to pay a constant dividend of$4 for the next 5 years with the first dividend in one year (t=1) and the last in 5 years (t=5).

The total required return is 10% pa.

What do you expected the share price to be in 5 years, just after the dividend at that time has been paid?

A European company just issued two bonds, a

• 1 year zero coupon bond at a yield of 8% pa, and a
• 2 year zero coupon bond at a yield of 10% pa.

What is the company's forward rate over the second year (from t=1 to t=2)? Give your answer as an effective annual rate, which is how the above bond yields are quoted.

An Australian company just issued two bonds:

• A 6-month zero coupon bond at a yield of 6% pa, and
• A 12 month zero coupon bond at a yield of 7% pa.

What is the company's forward rate from 6 to 12 months? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.

You deposit cash into your bank account. Have you or debt?

You deposit cash into your bank account. Does the deposit account represent a debt or to you?

Which of the following statements is NOT correct? Bond investors:

The coupon rate of a fixed annual-coupon bond is constant (always the same).

What can you say about the income return ($r_\text{income}$) of a fixed annual coupon bond? Remember that:

$$r_\text{total} = r_\text{income} + r_\text{capital}$$

$$r_\text{total, 0 to 1} = \frac{c_1}{p_0} + \frac{p_1-p_0}{p_0}$$

Assume that there is no change in the bond's total annual yield to maturity from when it is issued to when it matures.

Select the most correct statement.

From its date of issue until maturity, the income return of a fixed annual coupon:

You bought a house, primarily funded using a home loan from a bank. Which of the following statements is NOT correct?

Where can a private firm's market value of equity be found? It can be sourced from the company's:

Taking inflation into account when using the DDM can be hard. Which of the following formulas will NOT give a company's current stock price $(P_0)$? Assume that the annual dividend was just paid $(C_0)$, and the next dividend will be paid in one year $(C_1)$.

A real estate agent says that the price of a house in Sydney Australia is approximately equal to the gross weekly rent times 1000.

What type of valuation method is the real estate agent using?

Itau Unibanco is a major listed bank in Brazil with a market capitalisation of equity equal to BRL 85.744 billion, EPS of BRL 3.96 and 2.97 billion shares on issue.

Banco Bradesco is another major bank with total earnings of BRL 8.77 billion and 2.52 billion shares on issue.

Estimate Banco Bradesco's current share price using a price-earnings multiples approach assuming that Itau Unibanco is a comparable firm.

Note that BRL is the Brazilian Real, their currency. Figures sourced from Google Finance on the market close of the BVMF on 24/7/15.

Two years ago you entered into a fully amortising home loan with a principal of $1,000,000, an interest rate of 6% pa compounding monthly with a term of 25 years. Then interest rates suddenly fall to 4.5% pa (t=0), but you continue to pay the same monthly home loan payments as you did before. How long will it now take to pay off your home loan? Measure the time taken to pay off the home loan from the current time which is 2 years after the home loan was first entered into. Assume that the lower interest rate was given to you immediately after the loan repayment at the end of year 2, which was the 24th payment since the loan was granted. Also assume that rates were and are expected to remain constant. Value the following business project to manufacture a new product.  Project Data Project life 2 yrs Initial investment in equipment$6m Depreciation of equipment per year $3m Expected sale price of equipment at end of project$0.6m Unit sales per year 4m Sale price per unit $8 Variable cost per unit$5 Fixed costs per year, paid at the end of each year $1m Interest expense per year 0 Tax rate 30% Weighted average cost of capital after tax per annum 10% Notes 1. The firm's current assets and current liabilities are$3m and $2m respectively right now. This net working capital will not be used in this project, it will be used in other unrelated projects. Due to the project, current assets (mostly inventory) will grow by$2m initially (at t = 0), and then by $0.2m at the end of the first year (t=1). Current liabilities (mostly trade creditors) will increase by$0.1m at the end of the first year (t=1).
At the end of the project, the net working capital accumulated due to the project can be sold for the same price that it was bought.
2. The project cost $0.5m to research which was incurred one year ago. Assumptions • All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year. • All rates and cash flows are real. The inflation rate is 3% pa. • All rates are given as effective annual rates. • The business considering the project is run as a 'sole tradership' (run by an individual without a company) and is therefore eligible for a 50% capital gains tax discount when the equipment is sold, as permitted by the Australian Tax Office. What is the expected net present value (NPV) of the project? Five years ago ($t=-5$ years) you entered into an interest-only home loan with a principal of$500,000, an interest rate of 4.5% pa compounding monthly with a term of 25 years.

Then interest rates suddenly fall to 3% pa ($t=0$), but you continue to pay the same monthly home loan payments as you did before. Will your home loan be paid off by the end of its remaining term? If so, in how many years from now? Measure the time taken to pay off the home loan from the current time which is 5 years after the home loan was first entered into.

Assume that the lower interest rate was given to you immediately after the loan repayment at the end of year 5, which was the 60th payment since the loan was granted. Also assume that rates were and are expected to remain constant.

The phone company Optus have 2 mobile service plans on offer which both have the same amount of phone call, text message and internet data credit. Both plans have a contract length of 24 months and the monthly cost is payable in advance. The only difference between the two plans is that one is a:

• 'Bring Your Own' (BYO) mobile service plan, costing $80 per month. There is no phone included in this plan. The other plan is a: • 'Bundled' mobile service plan that comes with the latest smart phone, costing$100 per month. This plan includes the latest smart phone.

Neither plan has any additional payments at the start or end. Assume that the discount rate is 1% per month given as an effective monthly rate.

The only difference between the plans is the phone, so what is the implied cost of the phone as a present value? Given that the latest smart phone actually costs $600 to purchase outright from another retailer, should you commit to the BYO plan or the bundled plan? The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time. What is the Net Present Value (NPV) of the project?  Project Cash Flows Time (yrs) Cash flow ($) 0 -100 1 11 2 121

A project's NPV is positive. Select the most correct statement:

You're considering a business project which costs $11m now and is expected to pay a single cash flow of$11m in one year. So you pay $11m now, then one year later you receive$11m.

Assume that the initial $11m cost is funded using the your firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa. Which of the following statements about the net present value (NPV), internal rate of return (IRR) and payback period is NOT correct?  Project Data Project life 2 yrs Initial investment in equipment$600k Depreciation of equipment per year $250k Expected sale price of equipment at end of project$200k Revenue per job $12k Variable cost per job$4k Quantity of jobs per year 120 Fixed costs per year, paid at the end of each year $100k Interest expense in first year (at t=1)$16.091k Interest expense in second year (at t=2) $9.711k Tax rate 30% Government treasury bond yield 5% Bank loan debt yield 6% Levered cost of equity 12.5% Market portfolio return 10% Beta of assets 1.24 Beta of levered equity 1.5 Firm's and project's debt-to-equity ratio 25% Notes 1. The project will require an immediate purchase of$50k of inventory, which will all be sold at cost when the project ends. Current liabilities are negligible so they can be ignored.

Assumptions

• The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio. Note that interest expense is different in each year.
• Thousands are represented by 'k' (kilo).
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are nominal. The inflation rate is 2% pa.
• All rates are given as effective annual rates.
• The 50% capital gains tax discount is not available since the project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

A company issues a large amount of bonds to raise money for new projects of similar risk to the company's existing projects. The net present value (NPV) of the new projects is positive but small. Assume a classical tax system. Which statement is NOT correct?

One method for calculating a firm's free cash flow (FFCF, or CFFA) is to ignore interest expense. That is, pretend that interest expense $(IntExp)$ is zero:

\begin{aligned} FFCF &= (Rev - COGS - Depr - FC - IntExp)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp \\ &= (Rev - COGS - Depr - FC - 0)(1-t_c) + Depr - CapEx -\Delta NWC - 0\\ \end{aligned}
Does this annual FFCF with zero interest expense or the annual interest tax shield?

Question 99  capital structure, interest tax shield, Miller and Modigliani, trade off theory of capital structure

A firm changes its capital structure by issuing a large amount of debt and using the funds to repurchase shares. Its assets are unchanged.

Assume that:

• The firm and individual investors can borrow at the same rate and have the same tax rates.
• The firm's debt and shares are fairly priced and the shares are repurchased at the market price, not at a premium.
• There are no market frictions relating to debt such as asymmetric information or transaction costs.
• Shareholders wealth is measured in terms of utiliity. Shareholders are wealth-maximising and risk-averse. They have a preferred level of overall leverage. Before the firm's capital restructure all shareholders were optimally levered.

According to Miller and Modigliani's theory, which statement is correct?

What is the covariance of a variable X with itself?

The cov(X, X) or $\sigma_{X,X}$ equals:

Assets A, B, M and $r_f$ are shown on the graphs above. Asset M is the market portfolio and $r_f$ is the risk free yield on government bonds. Assume that investors can borrow and lend at the risk free rate. Which of the below statements is NOT correct?

A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.

In the last 5 minutes, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. The risk free rate was unchanged.

What do you think was the stock's historical return over the last 5 minutes, given as an effective 5 minute rate?

A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.

Over the last year, bad economic news was released showing a higher chance of recession. Over this time the share market fell by 1%. The risk free rate was unchanged.

What do you think was the stock's historical return over the last year, given as an effective annual rate?

A firm changes its capital structure by issuing a large amount of equity and using the funds to repay debt. Its assets are unchanged. Ignore interest tax shields.

According to the Capital Asset Pricing Model (CAPM), which statement is correct?

The CAPM can be used to find a business's expected opportunity cost of capital:

$$r_i=r_f+β_i (r_m-r_f)$$

What should be used as the risk free rate $r_f$?

 Project Data Project life 1 year Initial investment in equipment $8m Depreciation of equipment per year$8m Expected sale price of equipment at end of project 0 Unit sales per year 4m Sale price per unit $10 Variable cost per unit$5 Fixed costs per year, paid at the end of each year $2m Interest expense in first year (at t=1)$0.562m Corporate tax rate 30% Government treasury bond yield 5% Bank loan debt yield 9% Market portfolio return 10% Covariance of levered equity returns with market 0.32 Variance of market portfolio returns 0.16 Firm's and project's debt-to-equity ratio 50%

Notes

1. Due to the project, current assets will increase by $6m now (t=0) and fall by$6m at the end (t=1). Current liabilities will not be affected.

Assumptions

• The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
• Millions are represented by 'm'.
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates.
• The project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

You deposit money into a bank account. Which of the following statements about this deposit is NOT correct?

Which of the following statements about returns is NOT correct? A stock's:

The capital market line (CML) is shown in the graph below. The total standard deviation is denoted by σ and the expected return is μ. Assume that markets are efficient so all assets are fairly priced.

Which of the below statements is NOT correct?

Fred owns some BHP shares. He has calculated BHP’s monthly returns for each month in the past 30 years using this formula:

$$r_\text{t monthly}=\ln \left( \dfrac{P_t}{P_{t-1}} \right)$$

He then took the arithmetic average and found it to be 0.8% per month using this formula:

$$\bar{r}_\text{monthly}= \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( r_\text{t monthly} \right)} }{T} =0.008=0.8\% \text{ per month}$$

He also found the standard deviation of these monthly returns which was 15% per month:

$$\sigma_\text{monthly} = \dfrac{ \displaystyle\sum\limits_{t=1}^T{\left( \left( r_\text{t monthly} - \bar{r}_\text{monthly} \right)^2 \right)} }{T} =0.15=15\%\text{ per month}$$

Assume that the past historical average return is the true population average of future expected returns and the stock's returns calculated above $(r_\text{t monthly})$ are normally distributed. Which of the below statements about Fred’s BHP shares is NOT correct?

A company advertises an investment costing $1,000 which they say is under priced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Of the 15% pa total expected return, the dividend yield is expected to be 4% pa and the capital yield 11% pa. Assume that the company's statements are correct. What is the NPV of buying the investment if the 15% total return lasts for the next 100 years (t=0 to 100), then reverts to 10% after that time? Also, what is the NPV of the investment if the 15% return lasts forever? In both cases, assume that the required return of 10% remains constant, the dividends can only be re-invested at 10% pa and all returns are given as effective annual rates. The answer choices below are given in the same order (15% for 100 years, and 15% forever): A company advertises an investment costing$1,000 which they say is underpriced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Of the 15% pa total expected return, the dividend yield is expected to always be 7% pa and rest is the capital yield.

Assuming that the company's statements are correct, what is the NPV of buying the investment if the 15% total return lasts for the next 100 years (t=0 to 100), then reverts to 10% after that time? Also, what is the NPV of the investment if the 15% return lasts forever?

In both cases, assume that the required return of 10% remains constant, the dividends can only be re-invested at 10% pa and all returns are given as effective annual rates.

The answer choices below are given in the same order (15% for 100 years, and 15% forever):

The total return of any asset can be broken down in different ways. One possible way is to use the dividend discount model (or Gordon growth model):

$$p_0 = \frac{c_1}{r_\text{total}-r_\text{capital}}$$

Which, since $c_1/p_0$ is the income return ($r_\text{income}$), can be expressed as:

$$r_\text{total}=r_\text{income}+r_\text{capital}$$

So the total return of an asset is the income component plus the capital or price growth component.

Another way to break up total return is to use the Capital Asset Pricing Model:

$$r_\text{total}=r_\text{f}+β(r_\text{m}- r_\text{f})$$

$$r_\text{total}=r_\text{time value}+r_\text{risk premium}$$

So the risk free rate is the time value of money and the term $β(r_\text{m}- r_\text{f})$ is the compensation for taking on systematic risk.

Using the above theory and your general knowledge, which of the below equations, if any, are correct?

(I) $r_\text{income}=r_\text{time value}$

(II) $r_\text{income}=r_\text{risk premium}$

(III) $r_\text{capital}=r_\text{time value}$

(IV) $r_\text{capital}=r_\text{risk premium}$

(V) $r_\text{income}+r_\text{capital}=r_\text{time value}+r_\text{risk premium}$

Which of the equations are correct?

Mr Blue, Miss Red and Mrs Green are people with different utility functions.

Note that a fair gamble is a bet that has an expected value of zero, such as paying $0.50 to win$1 in a coin flip with heads or nothing if it lands tails. Fairly priced insurance is when the expected present value of the insurance premiums is equal to the expected loss from the disaster that the insurance protects against, such as the cost of rebuilding a home after a catastrophic fire.

Which of the following statements is NOT correct?

Mr Blue, Miss Red and Mrs Green are people with different utility functions. Which of the statements about the 3 utility functions is NOT correct?

Mr Blue, Miss Red and Mrs Green are people with different utility functions. Which of the statements about the 3 utility functions is NOT correct?

Mr Blue, Miss Red and Mrs Green are people with different utility functions.

Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?

Mr Blue, Miss Red and Mrs Green are people with different utility functions.

Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose$50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose$50. Which of the following statements is NOT correct?

Which of the following is NOT a valid method for estimating the beta of a company's stock? Assume that markets are efficient, a long history of past data is available, the stock possesses idiosyncratic and market risk. The variances and standard deviations below denote total risks.

An economy has only two investable assets: stocks and cash.

Stocks had a historical nominal average total return of negative two percent per annum (-2% pa) over the last 20 years. Stocks are liquid and actively traded. Stock returns are variable, they have risk.

Cash is riskless and has a nominal constant return of zero percent per annum (0% pa), which it had in the past and will have in the future. Cash can be kept safely at zero cost. Cash can be converted into shares and vice versa at zero cost.

The nominal total return of the shares over the next year is expected to be:

A person is thinking about borrowing $100 from the bank at 7% pa and investing it in shares with an expected return of 10% pa. One year later the person will sell the shares and pay back the loan in full. Both the loan and the shares are fairly priced. What is the Net Present Value (NPV) of this one year investment? Note that you are asked to find the present value ($V_0$), not the value in one year ($V_1$). A managed fund charges fees based on the amount of money that you keep with them. The fee is 2% of the start-of-year amount, but it is paid at the end of every year. This fee is charged regardless of whether the fund makes gains or losses on your money. The fund offers to invest your money in shares which have an expected return of 10% pa before fees. You are thinking of investing$100,000 in the fund and keeping it there for 40 years when you plan to retire.

What is the Net Present Value (NPV) of investing your money in the fund? Note that the question is not asking how much money you will have in 40 years, it is asking: what is the NPV of investing in the fund? Assume that:

• The fund has no private information.
• Markets are weak and semi-strong form efficient.
• The fund's transaction costs are negligible.
• The cost and trouble of investing your money in shares by yourself, without the managed fund, is negligible.

A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever.

All rates are given as nominal effective annual returns. Assume that:

• His forecast is true.
• Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true.
• Ignore all costs such as taxes, agent fees, maintenance and so on.
• All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property.
• The non-monetary benefits of owning real estate and renting remain constant.

Which one of the following statements is NOT correct? Over time:

Fundamentalists who analyse company financial reports and news announcements (but who don't have inside information) will make positive abnormal returns if:

Economic statistics released this morning were a surprise: they show a strong chance of consumer price inflation (CPI) reaching 5% pa over the next 2 years.

This is much higher than the previous forecast of 3% pa.

A vanilla fixed-coupon 2-year risk-free government bond was issued at par this morning, just before the economic news was released.

What is the expected change in bond price after the economic news this morning, and in the next 2 years? Assume that:

• Inflation remains at 5% over the next 2 years.
• Investors demand a constant real bond yield.
• The bond price falls by the (after-tax) value of the coupon the night before the ex-coupon date, as in real life.

A man inherits $500,000 worth of shares. He believes that by learning the secrets of trading, keeping up with the financial news and doing complex trend analysis with charts that he can quit his job and become a self-employed day trader in the equities markets. What is the expected gain from doing this over the first year? Measure the net gain in wealth received at the end of this first year due to the decision to become a day trader. Assume the following: • He earns$60,000 pa in his current job, paid in a lump sum at the end of each year.
• He enjoys examining share price graphs and day trading just as much as he enjoys his current job.
• Stock markets are weak form and semi-strong form efficient.
• He has no inside information.
• He makes 1 trade every day and there are 250 trading days in the year. Trading costs are $20 per trade. His broker invoices him for the trading costs at the end of the year. • The shares that he currently owns and the shares that he intends to trade have the same level of systematic risk as the market portfolio. • The market portfolio's expected return is 10% pa. Measure the net gain over the first year as an expected wealth increase at the end of the year. In late 2003 the listed bank ANZ announced a 2-for-11 rights issue to fund the takeover of New Zealand bank NBNZ. Below is the chronology of events: • 23/10/2003. Share price closes at$18.30.

• 24/10/2003. 2-for-11 rights issue announced at a subscription price of $13. The proceeds of the rights issue will be used to acquire New Zealand bank NBNZ. Trading halt announced in morning before market opens. • 28/10/2003. Trading halt lifted. Last (and only) day that shares trade cum-rights. Share price opens at$18.00 and closes at $18.14. • 29/10/2003. Shares trade ex-rights. All things remaining equal, what would you expect ANZ's stock price to open at on the first day that it trades ex-rights (29/10/2003)? Ignore the time value of money since time is negligibly short. Also ignore taxes. A company announces that it will pay a dividend, as the market expected. The company's shares trade on the stock exchange which is open from 10am in the morning to 4pm in the afternoon each weekday. When would the share price be expected to fall by the amount of the dividend? Ignore taxes. The share price is expected to fall during the: Currently, a mining company has a share price of$6 and pays constant annual dividends of $0.50. The next dividend will be paid in 1 year. Suddenly and unexpectedly the mining company announces that due to higher than expected profits, all of these windfall profits will be paid as a special dividend of$0.30 in 1 year.

If investors believe that the windfall profits and dividend is a one-off event, what will be the new share price? If investors believe that the additional dividend is actually permanent and will continue to be paid, what will be the new share price? Assume that the required return on equity is unchanged. Choose from the following, where the first share price includes the one-off increase in earnings and dividends for the first year only $(P_\text{0 one-off})$ , and the second assumes that the increase is permanent $(P_\text{0 permanent})$:

Note: When a firm makes excess profits they sometimes pay them out as special dividends. Special dividends are just like ordinary dividends but they are one-off and investors do not expect them to continue, unlike ordinary dividends which are expected to persist.

format: plain


text:
# Data and Plots

## Datasets

R comes with many datasets built in. These are part of the datasets package that is always loaded in R. For example, the mtcars dataset is a well-known dataset from Motor Trend magazine, documenting fuel consumption and vehicle characteristics for a number of vehicles. At the R console, typing mtcars will print the entire dataset.

You can find help on datasets as usual using the Help tab in RStudio, clicking on the Packages link and navigating to the datasets package.

## Import data

To do any real work, one has to load data from an external source. RStudio makes it easy to import data.

Consider the data set that will be used in Lab 2, which is the 100m times for men and women. We will illustrate importing this data set, step by step.

### Step 1

From the Import Dataset menu, select From CSV to get a dialog as shown below and navigate to the folder containing the 100men file.

Note that the import dialog has a number of options and on the right buttom it shows a preview of the code that will be used to import the data. If one cut and pasted the code into the R console, the result would be the same as what one would get via the dialogs.

RStudio also take care to name the variable that will hold data according R conventions using X100men!

### Step 2

When you open the file, RStudio shows a preview of the data in the viewer window.

This is of course not what we want since a cursory inspection shows that the data appears to contain three columns. So obviously, we have specified something wrong.

### Step 3

In the Import Options panel, change the delimeter to Tab and while we are at it, change the name to data.men. Notice how the code preview reflects changes made to these options.

### Step 4

Press the Import button to get the data into R.

The result of the import is a variable called data.men that contains the data. Data formatted this way (either tab-delimeted, or comma-separated, or spread-sheet like) is so common that R has a abstraction for it: the data frame. You will have more opportunity to learn about data frames in the data parts of the course.

### Avoiding dialogs

As one becomes more and more familiar with R, direct code becomes preferable to the slower interactive dialogs. This is one reason that RStudio gives you the code preview, to aid in your learning process. So, to get the same effect as the above dialog process did, one could have pasted the RStudio code into an R console to get the same result.

library(readr)
data.men <- read_delim("100men", "\t", escape_double = FALSE, trim_ws = TRUE)
## Parsed with column specification:
## cols(
##   Athlete = col_character(),
##   Time = col_double(),
##   Date = col_character()
## )

That would create the same data set.

With more complex structures like data frames, the function str (for structure) is a good way to examine them.

str(data.men)
## Classes 'tbl_df', 'tbl' and 'data.frame':    20 obs. of  3 variables:
##  $Athlete: chr "Usain Bolt (Jamaica)" "Usain Bolt (Jamaica)" "Usain Bolt (Jamaica)" "Asafa Powell (Jamaica)" ... ##$ Time   : num  9.58 9.69 9.72 9.74 9.77 9.79 9.84 9.85 9.86 9.9 ...
##  $Date : chr "Aug 16, 2009" "Aug 16, 2008" "May 31, 2008" "Sept 9, 2007" ... ## - attr(*, "spec")=List of 2 ## ..$ cols   :List of 3
##   .. ..$Athlete: list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## .. ..$ Time   : list()
##   .. .. ..- attr(*, "class")= chr  "collector_double" "collector"
##   .. ..$Date : list() ## .. .. ..- attr(*, "class")= chr "collector_character" "collector" ## ..$ default: list()
##   .. ..- attr(*, "class")= chr  "collector_guess" "collector"
##   ..- attr(*, "class")= chr "col_spec"

We see that the data consists of 20 observations on 3 variables: Athlete, Time, Date. The second is numeric while the others are character.

### More on data import

RStudio provides ways to import data directly from spreadsheets like Excel, etc. You can explore these options on your own.

RStudio makes use of some packages to import data, notably the readr package. Strictly speaking these packages are not necessary for the job, but such packages include improvements that make them attractive. For example, a vanilla installation of R provides functions like read.csv and read.delim (analogous to read_csv, read_delim) that can also be used. However, by default, these functions perform some conversions, treating character variables as factors, for example. That can be troublesome (and computationally expensive) when dealing with large data sets. In this class, some instructors may use these vanilla R functions with various options to control the behavior.

## Graphs and Plots

Graphing/plotting are among the great strengths of R. There are two main main approaches that are common in building graphs and plots.

1. Using basic functions provided by R itself via the graphics package which has a number of standard facilities. A quick way to familiarize yourself with base graphics is to type the command demo(graphics) at the R console to see its capabilities.

2. Using a package like ggplot2, which requires a more nuanced understanding of a graphics object. You will have to install this package. ggplot2 implements a grammar of graphics and so takes a bit more work to use, but is quite powerful.

Both approaches allow for step-by-step building up of complex plots, and creating PDFs or images that can be included in other documents. Although ggplot2 is becoming more popular, many packages may not use ggplot2 for plotting. Furthermore, some special plots created by packages may use one of base graphics or ggplot2 and so there isn’t a ready made equivalent in the other, although it can be constructed with extra work. So you will see both bae graphics and ggplot2 used in this course.

For ease of use, ggplot2 provides a function called qplot that can emulate the base graphics plot function capabilities. This offers a quick way to begin using ggplot2, initially.

Description Base Graphics ggplot2
Plot y versus x using points plot(x, y) qplot(x, y)
Plot y versus x using lines plot(x, y, type = "l") qplot(x, |y, geom = "line")
Plot y versus x using both points and lines plot(x, y, type = |"b") qplot(x, y, geom = c("point", "line"))
Boxplot of x boxplot(x) qplot(x, geom = "boxplot")
Side-by-side boxplot of x and y boxplot(x, y) qplot(x, y, geom = "boxplot")
Histogram of x hist(x) qplot(x, geom = "histogram")

#### Examples

It is a good idea to try out the functions using the example function. At the R console type,

example(plot)

to see the plot examples.

For ggplot2, you will have to load the library first and then use example.

library(ggplot2)
example(qplot)


text:
# Is the angular velocity of a wave constant?

For a given wave, how does the angular frequency vary? For example, if the wave starts propagating at $$x=0$$ with angular frequency $$\omega$$, then at another point in the $$x$$ direction, does this angular velocity remain same ?

• Hi and welcome to physics.SE! What sort of "wave" are you considering here? What is your definition of "angular velocity" of this wave? Also, please consider writing more useful question titles, see How do we write good question titles? – ACuriousMind Jan 6 at 13:50

$$\omega=2\pi f$$ so angular frequency depends only on frequency. And frequency depends only on the source that produces it, that is, it is not a function of time or position. So angular frequency is constant.

• Hi, welcome to PhysicsSE! Please use TeX for formulas. – Mauro Giliberti Jan 6 at 15:12


text:
Implementing std::format

One of the most exciting new functions in C++20 is std::format. I have never liked the ways that strings are formatted in C++ and have been guilty many times of using sprintf and such.

Since std::format is not implemented by any of the major compilers at this time, I thought I would try out my own very basic implementation and this is what I have come up with:

#include <string_view>
#include <iostream>
#include <string>
#include <sstream>
#include <stdexcept>

void format_helper(std::ostringstream& oss, std::string_view str) // base function
{
oss << str;
}

template<typename T, typename... Targs>
void format_helper(std::ostringstream& oss, std::string_view str, T value, Targs... args)
{
std::size_t openBracket = str.find_first_of('{');
if (openBracket != std::string::npos)
{
std::size_t closeBracket = str.find_first_of('}', openBracket + 1);
if (closeBracket == std::string::npos)
throw std::runtime_error("missing closing bracket.");
oss << str.substr(0, openBracket);
oss << value;
format_helper(oss, str.substr(closeBracket + 1), args...);
return;
}
oss << str;
}

std::string format(std::string_view str)
{
return std::string(str);
}

template<typename T, typename... Targs>
std::string format(std::string_view str, T value, Targs...args)
{
std::ostringstream oss;
format_helper(oss, str, value, args...);
return oss.str();
}

int main()
{
int a = 5;
double b = 3.14;
std::string c("hello");

// correct number of arguments
std::cout << format("a = {}, b = {}, c = {}", a, b, c) << "\n";

// too few arguments
std::cout << format("a = {}, b = {}, c = {}", a, b) << "\n";

// too many arguments
std::cout << format("a = {}, b = {}, c = {}", a, b, c, 12.4) << "\n";
}

• Funny how you didn't put the \n in the format string :) Oct 27, 2021 at 16:51
• @G.Sliepen yeah that is a bit ironic :-)
– jdt
Oct 27, 2021 at 17:44

Library includes are almost sorted, but <string_view> is out of place - perhaps an oversight?

We can use a fold-expression to avoid having to write specific recursion in format_helper() (yes, I know that's usually not a recursive call, just template expansion). See below.

format_helper() should be in an implementation namespace, where it's less visible to client code.

I wouldn't throw an exception for the unterminated {} in the format string. At first glance, this looks like it's a programming error, but consider that in a production program, the format string probably comes from a translation database. It's probably more useful to translators to see the unreplaced part of the string than to get a much more vague exception message (that doesn't even indicate which format string caused it).

There's no reason to use find_first_of(char) rather than the shorter find(char). In fact, I'm not sure why the latter is provided.

Fold-expression version

From C++17 onwards, we have fold expressions, which make variadic templates much, much simpler to write and to reason about. For us, it means only one version of each function.

The key here is that we want to pass the string view by reference, so that printing each argument adjusts the format string seen by the remaining ones.

template<typename T>
void format_helper(std::ostringstream& oss,
std::string_view& str, const T& value)
{
std::size_t openBracket = str.find('{');
if (openBracket == std::string::npos) { return; }
std::size_t closeBracket = str.find('}', openBracket + 1);
if (closeBracket == std::string::npos) { return; }
oss << str.substr(0, openBracket) << value;
str = str.substr(closeBracket + 1);
}

template<typename... Targs>
std::string format(std::string_view str, Targs...args)
{
std::ostringstream oss;
(format_helper(oss, str, args),...);
oss << str;
return oss.str();
}

At this point, we could (arguably should) bring the helper inside format as a named lambda, so we define only a single function.

• If you just use it to find a single char, then find_first_of() would indeed be redundant, but if you pass it a string then find_first_of() and find() do something different. Oct 27, 2021 at 16:34
• @G.Sliepen: yes, that's why I was specific about the overload that takes a char. Oct 27, 2021 at 19:10
• Symmetry I guess? It would be weird if it accepts everything that find() does except a char. But find() indeed makes more sense here. Oct 27, 2021 at 21:06
• Thanks for your review, especially re. fold expressions! Is there any benefit over using a named lambda over simply static void format_helper?
– jdt
Oct 28, 2021 at 13:05
• One benefit is almost the same as we could get by writing namespace format_detail { template<⋯> void format_helper(⋯); }: it's not polluting the client code's namespace (remember that, being a template, the name must be visible in every TU that instantiates format(); we don't gain much from static linkage). The other benefit, and this one is more subjective, is that it's declared right by its only use, so everything is in one place in the source. That's only a slight benefit, especially when the source file comprises just those two functions. Oct 28, 2021 at 13:14

If you pass a string with no arguments, it will not do any checking for embedded control sequences and will print the string as-is without any error and without handling escaped out sequences.

You should break out the brace finding into separate functions to make it easier to extend, such as adding "{{" to mean a literal open-brace character, and make the contents inside the braces available as formatting options.

The error should mention "}" rather than just "closing bracket" since someone reading this error might not realize you're referring to the curly brace. For that matter, it should indicate the fact that the error comes from the formatting code, and ought to show the bad string.

You should use a single character rather than a nul-terminated character string when you only have a single character. cout << '\n'; will be more efficient and generate less code than using "\n".

All in all, it's good and simple. Something like this was an early example for the power for variadic templates and how that could replace the ostream paradigm. For your own stuff, you don't have to make it look somewhat similar to the fmt library, but could use your own (simple) conventions for indicating replacements and how they can themselves be escaped out.


text:
S.V.Kerov and N.V.Tsilevich.
Stick breaking process generates virtual permutations with Ewens distribution.
Zapiski Nauchn. Semin. POMI, 223 (1995), 162-180.
English translation: J. Math. Sci. (N.Y.), 87, No. 6 (1997), 4082-4093.

Abstract. Given a sequence $x$ of points in the unit interval, we associate with it a virtual permutation $w=w(x)$ (that is, a sequence $w$ of permutations $w_n \in \frak S_n$ such that $w_{n-1}=w_n'$ is obtained from $w_n$ by erasing the last element $n$ from its cycle, for all $n=1,2,\,\ldots$). We introduce a detailed version of the well-known stick breaking process, generating a random sequence $x$. It is proved that the associated random virtual permutation $w(x)$ has Ewens' distribution. Up to subsets of zero measure, the space $\frak S^\infty = \varprojlim \frak S_n$ of virtual permutations is identified with the cube $[0,1]^\infty$.

Back to the list of papers