# 2022 Australian National University|EMET3007/8012 Assignment 2

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## Instructions:

This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

# 2022 Australian National University|EMET3007/8012 Assignment 2

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## Instructions:

This assignment is worth either $20 \%$ or $25 \%$ of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by $5 \mathrm{pm}$ Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck.

[10 marks] In this exercise we will consider four different specifications for forecasting monthly Australian total employed persons. The dataset (available on Wattle) AUSEmp 1oy 2022. csv contains three columns; the first column contains the date; the second contains the sales figures for that month (FRED data series LFEMTTTTAUM647N), and the third contains Australian GDP for that month.1] The data runs from January 1995 to January $2022 .$

Let $M_{i t}$ be a dummy variable that denotes the month of the year. Let $D_{i t}$ be a dummy variable which denotes the quarter of the year. The four specifications we consider are
\begin{aligned} &S_1: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\epsilon_t \ &S_2: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\epsilon_t \ &S_3: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\epsilon_t \ &S_4: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$.

a) For each specification, describe this specification in words.
b) For each specification, estimate the values of the parameters, and compute the MSE, $\mathrm{AIC}$, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code.
c) For each specification, compute the MSFE for the 1-step and 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$.
d) For each specification, plot the out-of-sample forecasts and comment on the results.

[10 marks] Now add to Question 1 the additional assumption that $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$. One estimator ${ }^2$ for $\sigma^2$ is
$$\hat{\sigma}^2=\frac{1}{T-k} \sum_{t=1}^T\left(y_t-\hat{y}_t\right)^2$$
where $\hat{y}_t$ is the estimated value of $y_t$ in the model and $k$ is the number of regressors in the specification.
a) For each specification $\left(S_1, \ldots, S_4\right)$, compute $\hat{\sigma}^2$.
b) For each specification, make a $95 \%$ probability forecast for the sales in June $2021 .$
c) For each specification, compute the probability that the total employed persons in June 2022 will be greater than $13.5$ million. According to the FRED series LFEMTTTTAUM647N, what was the actual employment level for that month.
d) Do you think the assumption that $\epsilon_t$ is iid is a reasonable assumption for this data series.

[10 marks] Here we investigate whether adding GDP $\mathrm{Gs}^3$ as a predictor can improve our forecasts. Consider the following modified specifications:
\begin{aligned} &S_1^{\prime}: y_t=a_0+a_1 t+\alpha_4 D_{4 t}+\gamma x_{t-h}+\epsilon_t \ &S_2^{\prime}: y_t=a_1 t+\sum_{i=1}^4 \alpha_i D_{i t}+\gamma x_{t-h}+\epsilon_t \ &S_3^{\prime}: y_t=a_0+a_1 t+\beta_{12} M_{12, t}+\gamma x_{t-h}+\epsilon_t \ &S_4^{\prime}: y_t=a_1 t+\sum_{i=1}^{12} \beta_i M_{i t}+\gamma x_{t-h}+\epsilon_t \end{aligned}
where $\mathbb{E} \epsilon_t=0$ for all $t$, and $x_{t-h}$ is GDP at time $t-h$. For each specification, compute the MSFE for the 1-step ahead, and the 5-step ahead forecasts, with the out-of-sample forecasting exercise beginning at $T_0=50$. For each specification, plot the out-of-sample forecasts and comment on the results.

[15 marks] Here we investigate whether Holt-Winters smoothing can improve our forecasts. Use a Holt-Winters smoothing method with seasonality, to produce 1-step ahead and 5-step ahead forecasts and compute the MSFE for these forecasts. You should use smoothing parameters $\alpha=\beta=\gamma=0.3$ and start the out-of-sample forecasting exercise at $T_0=50$. Plot these out-of-sample forecasts and comment on the results.
Additionally, estimate the values for $\alpha, \beta$, and $\gamma$ which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline $\alpha=\beta=\gamma=0.3$.

[5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Total Employment. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs).

[10 marks] Develop another model, either based on material from class or otherwise, to forecast Australian Total Employment. Your new model should perform better (have a lower MSFE or MAFE) than all models from Questions 1,3, and 4. As part of your response to this question you must provide:
a) a brief written explanation of what your model is doing,
b) a brief statement on why you think your new model will perform better,
c) any relevant equations or mathematics/statistics to describe the model,
d) the code to run the model, and
e) the MSFE and/or MAFE error found by your model, and a brief discussion of how this compares to previous cases.

[15 marks] Consider the ARX(1) model
$$y_t=\mu+a t+\rho y_{t-1}+\epsilon_t$$
where the errors follow an $\mathrm{AR}(2)$ process
$$\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad \mathbf{u} \sim \mathcal{N}\left(0, \sigma^2 I\right)$$
for $t=1, \ldots, T$ and $e_{-1}=e_0=0$. Suppose $\phi_1, \phi_2$ are known. Find (analytically) the maximum likelihood estimators for $\mu, a, \rho$, and $\sigma^2$.

Hint: First write $y$ and $\epsilon$ in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of $\epsilon$ and $y$. Then apply some appropriate calculus. You may want to let $H=I-\phi_1 L-\phi_2 L^2$, where $I$ is the $T \times T$ identity matrix, and $L$ is the lag matrix.

# 液体和等离子体|PHYS3202 Fluids and Plasma代写

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“Fluids and Plasmas” is an applied physics course designed to give the students experience in working with, predicting and measuring the behaviour of fluid flows and plasmas. The course begins with an outline of the fluid equations of motion, which lead to solutions for waves in fluids, convection and buoyancy-driven flows.

Here we consider some special cases of $(100)$ obtained by specializing $a, b, c$, and $d$ in $H$ of (77). Our choices for these four functions will determine the structure of the first integrals $x_{0}$ and $y_{0}$ through (99). For the cases we consider, their structure will be easy to discern and will give some insight into the behavior of $\xi$. How $\mathbf{B}_{p}$ will propagate in each case is pointed out to make the discussion more physically concrete. To conclude, a physical interpretation for the terms of $H$ and the role they play in determining how solutions propagate are discussed as well.

The first case we consider is a rather drastic simplification of the general result (99): we take $a, b, c$, and $d$ all to be zero, getting rid of Hentirely. Then we are simply left with
$$x_{0}=x \quad \text { and } \quad y_{0}=y .$$
Thus, in this case, the general solution for $\xi$ is of the form
$$\xi=\xi(x, y, z-\gamma \tau),$$
which corresponds to a structure propagating toroidally with specd $\gamma$.

$$\psi=(1 / \gamma)\left(\xi-\alpha \nabla_{1}^{2} \xi\right)(x, y, z,-\gamma \tau) .$$
The arguments in parentheses stress that $\psi$ moves in exactly the same way as $\xi$ : surfaces of constant poloidal flux simply propagate in the $z$ direction with constant velocity $\gamma$. Applying $\mathbf{B}{p}=-\epsilon B{T} \hat{\mathbf{z}} \times \nabla_{1} \psi$ to (105) shows that the disturbance $\mathbf{B}{p}$ also propagates in the same way: if we follow a point moving along a characteristic curve, $\mathbf{B}{p}$ at the point will be a constant vector. However, from $(105)$ and the arguments given at the end of Sec. III F, the solution is not necessarily an Alfvén-like wave because, in general, $\mathbf{B}{p}$ will not be proportional to $\mathbf{v}{t}$ for this case.

## PHYS3202 COURSE NOTES ：

Having introduced the fluid equations, we next discuss a method for arriving at exact solutions of them.

We denote the partial derivative of a quantity by a subscript, e.g., $\partial U / \partial \tau \equiv U_{\tau}$. Then, after rearranging the terms of $(9)$ and $(10)$ and subtracting (14) from (9), we can write
\begin{aligned} &U_{r}+[\phi, U]+J_{z}+[J, \psi]=0, \ &\psi_{r}+(\phi-\alpha \chi){z}+[\phi-\alpha \chi, \psi]=0, \end{aligned} This is the nonlinear system we will study. Note that we are taking $\hat{\eta}=0$ in (16); the resistivity of the plasma is neglected for all that follows. To satisfy (17) we take $$\chi=g(z)+U,$$ where $g$ is an arbitrary function of $z$. This is by no means the general solution to (17); it is simply a special case that satisfies (17) with little effort. Defining \begin{aligned} &\xi \xi \phi-\alpha g(z), \ &\text { and recasting (15) and (16) in terms of } \xi \text { gives } \ &\qquad U{+}+[\xi, U]+J_{z}+[J, \psi]=0 \ &\text { and } \ &\qquad \psi_{\tau}+(\xi-\alpha U){z}+[\xi-\alpha U, \psi]=0, \end{aligned} where (18) has been used. We note in passing that from (19) and (6), the definition of $U$, we have $$U=\nabla{1}^{2} \xi,$$
a relation that will be used often in what follows.
Now we have to find solutions to (20) and (21). Let us first consider the simpler case of axisymmetric equilibrium.

# 物理基础1A| Physics 1A: Foundations PHYS08016代写

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This is an introductory-level course, covering the classical physics of kinematics, dynamics, oscillations, forces and fields, and touching on aspects of contemporary physics, including relativity and chaos. The course is designed for those with qualifications in physics and mathematics at SCE-H level or equivalent.

During an IFR flight, a passenger looks out the window while relaxing in his seat. He observes a turn and estimates the bank angle to be $30^{\circ}$. At the same time, the passenger observes the free surface of the orange juice in his glass: it is parallel to the tray.
a) The passenger assumes the turn being flown as a rate one turn. Explain the term rate one turn. Why is it correct to assume a rate one turn.
b) The passenger assumes the turn being flown as a coordinated turn. Explain the term coordinated turn. Why is it correct to assume a coordinated turn.
c) Calculate the aircraft’s true airspeed.

a) A rate one turn is a turn with a heading change of $180^{\circ}$ in 60 seconds. During IFR flights turns are performed as rate one turns.
b) In a coordinated turn (correctly banked turn)

• the lift force lies in the aircraft plane of symmetry,
• the ball in the turn and slip indicator is centered,
• there is no acceleration along the $y$-axis of the aircraft.
This phenomenon is also shown by the free surface of the orange juice in the glass which is parallel to the tray.
c) $\tan \Phi=\frac{V \cdot \Omega}{g}$
$$V=\frac{g}{\Omega} \cdot \tan \Phi=\frac{9.81 \mathrm{~m} \cdot 60 \mathrm{~s}}{\mathrm{~s}^{2} \cdot \pi} \cdot \tan 30^{\circ}=108.2 \frac{\mathrm{m}}{\mathrm{s}}=210 \mathrm{kt}$$
Answer: The aircraft’s true airspeed is $210 \mathrm{kt}$.

## FINM2001 COURSE NOTES ：

The category of effect of a failure is judged to be hazardous. Following $A C J N o$. 1 . to $J A R$ $25.1309$
a) What is the largest permissible failure probability?
b) What is the mean time to failure $M T T F$ ?
Solution
$F(t) \quad$ probability of failure,
$\lambda \quad$ failure rate
MTTF mean time to failure
FH flight hour
a) hazardous : $F(t=1 \mathrm{FH}) \leq 10^{-7}$
b) For small probabilities of failure: $\quad \lambda \approx F / t=10^{-7} \cdot \frac{1}{\mathrm{FH}}$
$$\mathrm{MTTF}=1 / \lambda=\frac{1}{10^{-7}} \mathrm{FH}=10000000 \mathrm{FH}$$
Answer: If a failure has a hazardous effect, the mean time to this failure may not be less than $10000000 \mathrm{FH}$.

# 公司财务|FINM2001 Corporate Finance代写

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This course focuses on tools and techniques used in modern financial management. Material in the course has an applied focus and is designed to provide students with the knowledge and skills required for understanding, exploring and analysing financial management issues. The course draws upon topical material in order to contextualise theoretical discussion, and present students with examples in practice.

Throughout the late 1950 ‘s, Myron J. Gordon (initially working with Ezra Shapiro) formalised the impact of distribution policies and their associated returns on current share price using the derivation of a constant growth formula, the mathematics for which are fully explained in the CVT text.

Required:

1. Present a mathematical summary of the Gordon Growth Model under conditions of certainty.
2. Comment on its hypothetical implications for corporate management seeking to maximise shareholder wealth.

These questions not only provide an opportunity to test your understanding of the companion text, but also to practise your written skills and ability to editorialise source material.

1. The Gordon Model
According to Gordon (1962) movements in ex-div share price $\left(\mathrm{P}_{2}\right.$ ) under conditions of certainty relate to the profitability of corporate investment and not dividend policy.

Using Gordon’s original notation and our Equation numbering from $C V T$ (Chapter Three) where $\mathrm{K}{e}$ represents the equity capitalisation rate; $E{1}$ equals next year’s post-tax earnings; $b$ is the proportion retained; (1-b) $E_{1}$ is next year’s dividend; $r$ is the return on reinvestment and r.b equals the constant annual growth in dividends:
(16) $\quad \mathrm{P}{0}=(1-\mathrm{b}) \mathrm{E}{1} / \mathrm{K}{\varepsilon}-\mathrm{rb} \quad$ subject to the proviso that $\mathrm{K}{\varepsilon}>$ r.b for share price to be finite.
You will also recall that in many Finance texts today, the equation’s notation is simplified with $D_{1}$ and g representing the dividend term and growth rate, subject to the constraint that $\mathrm{K}{\varepsilon}>\mathrm{g}$ (17) $\mathrm{P}{\mathrm{b}}=\mathrm{D}{1} / \mathrm{K}{\mathrm{c}}-\mathrm{g}$

1. The Implications
In a world of certainty, Gordon’s analysis of share price behaviour confirms the importance of Fisher’s relationship between a company’s return on reinvestment $(\mathrm{r})$ and its shareholders’ opportunity cost of capital rate $(\mathrm{K})$ ).

## FINM2001 COURSE NOTES ：

Moving into a world of uncertainty, Gordon (op cit) explains why rational-risk averse investors are no longer indifferent to managerial decisions to pay a dividend or reinvest earnings on their behalf, which therefore impacts on share price.

Required:

1. Present a mathematical summary of the difference between the Gordon Growth Model under conditions of certainty and uncertainty.
2. Comment on its hypothetical implications for corporate management seeking to maximise shareholder wealth.
An Indicative Outline Solution
Again, these questions provide opportunities to test your understanding of the companion text and practise your written and editorial skills.
3. The Gordon Model and Uncertainty
According to Gordon (ibid) movements in share price under conditions of uncertainty relate to dividend policy, rather than investment policy and the profitability of corporate investment. He begins with the basic mathematical growth model:
(16) $\mathrm{P}{0}=(1-\mathrm{b}) \mathrm{E}{1} / \mathrm{K}{c}-\mathrm{rb} \quad$ subject to the proviso that $\mathrm{K}{\varepsilon}>$ r.b for share price to be finite.
This again simplifies to:
(17) $\mathrm{P}{0}=\mathrm{D}{1} / \mathrm{K}{c}-\mathrm{g}$ subject to the constraint that $\mathrm{K}{c}>\mathrm{g}$
But now, the overall shareholder return (equity capitalisation rate) is no longer a constant but a function of the timing and size of the dividend payout. Moreover, an increase in the retention ratio also results in a further rise in the periodic capitalisation rate. Expressed mathematically:
$$\mathrm{K}{\varepsilon}=f\left(\mathrm{~K}{\mathrm{el}}<\mathrm{K}{e 2}<\ldots \mathrm{K}{\mathrm{en}}\right)$$
4. The Implications
According to Gordon’s uncertainty hypothesis, rational, risk averse investors adopt a “bird in the hand” philosophy to compensate for the non-payment of future dividends.

They prefer dividends now, rather than later, even if retentions are more profitable than distributions (i.e. $r>\mathrm{K}{\mathrm{e}}$ ). They prefer high dividends to low dividends period by period. (i.e. $\mathrm{D}{1}>\mathrm{D}{2}$ ). . Near dividends and higher payouts are discounted at a lower rate ( $\mathrm{K}{\mathrm{ct}}$ now dated) ,
Thus, investors require a higher overall average return on equity $\left(\mathrm{K}_{c}\right.$ ) from firms that retain a higher proportion of earnings with obvious implications for share price. It will fall.

# 金融学基础|FINM1001 Foundations of Finance代写

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This course is designed to familiarise students with the components of the financial system as well as to introduce them to the three basic ideas underpinning finance, namely the time value of money, diversification and arbitrage. In doing so, the course provides students with introductory exposure to financial transactions, institutions and markets including money markets, stock markets, foreign exchange and derivative markets and the instruments traded therein. It also provides students with a solid foundation for later studies in finance.

If you wish to provide 520,000 for your newborn’s University education, how much should you imvest now, given the interest rate that will accrue on the investment is $10 \%$ p.a compounded monthly?

In order to determine how much you should invest now, calculate the present value of $\$ 20,000received 18 years from now, bearing in mind that interest is compounded monthly. \begin{aligned} P V &=\frac{\ 20,000}{\left(1+\frac{0.10}{12}\right)^{12_{x} 13}} \ &=\ 3,330.73 \end{aligned} ## FINM1001 COURSE NOTES ： A company needs\$10,000$ in 5 years to replace a piece of equipment. How much must be invested each year at $8 \%$ p.a compounded semi-annualy in order to provide for this replacement?

To determine the amount the company must invest annually, simply use the future value of an annuity formula, bearing in mind that the interest rate is compounded semi-annually. Therefore, as investments will be made on an annual basis, we must calculate an annual effective interest rate to use in the annuity calculation.
\begin{aligned} r &=\left(1+\frac{0.08}{2}\right)^{2}-1 \ &=0.0816 \ &=8.16 \% \end{aligned}
\begin{aligned} \ 10,000 &=F\left[\frac{(1.0816)^{3}-1}{0.0816}\right] \ F &=\frac{\ 10,000}{\left[\frac{(1.0816)^{5}-1}{0.0816}\right]} \ &=\ 1,699.14 \end{aligned}

# 科学计算 Scientific Computing MATH3511/MATH6111

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To be more specific, suppose that
$$F(h)=a_{0}+a_{1} h^{p}+\mathcal{O}\left(h^{r}\right)$$
as $h \rightarrow 0$ for some $p$ and $r$, with $r>p$. We assume that we know the values of $p$ and $r$, but not $a_{0}$ or $a_{1}$. Indeed, $F(0)=a_{0}$ is the quantity we seek. Suppose that we have computed $F$ for two stepsizes, say, $h$ and $q h$ for some $q>1$. Then we have
$$F(h)=a_{0}+a_{1} h^{p}+\mathcal{O}\left(h^{r}\right)$$
and
$$F(q h)=a_{0}+a_{1}(q h)^{p}+\mathcal{O}\left(h^{r}\right) .$$
This system of two linear equations in the two unknowns $a_{0}$ and $a_{1}$ is easily solved to obtain
$$a_{0}=F(h)+\frac{F(h)-F(q h)}{q^{p}-1}+\mathcal{O}\left(h^{r}\right) .$$
Thus, the accuracy of the improved value, $a_{0}$, is $\mathcal{O}\left(h^{r}\right)$ rather than only $\mathcal{O}\left(h^{p}\right)$.

## MATH3511/MATH6111COURSE NOTES ：

For example, applying Euler’s method to this equation using a fixed stepsize $h$, we have
$$y_{k+1}=y_{k}+\lambda y_{k} h=(1+\lambda h) y_{k},$$
which means that
$$y_{k}=(1+\lambda h)^{k} y_{0} .$$
Provided $\lambda<0$, the exact solution decays to zero as $t$ increases, as will the computed solution if $|1+\lambda h|<1$. This result agrees with our earlier stability analysis because $J=\lambda$ for this ODE. We also note that the growth factor $1+\lambda h$ agrees with the series expansion
$$e^{\lambda h}=1+\lambda h+\frac{(\lambda h)^{2}}{2}+\frac{(\lambda h)^{3}}{6}+\cdots$$
through terms of first order in $h$, and hence Euler’s method is first-order accurate. Especially for more complicated numerical methods, a linear ODE is easier to work with than a general ODE, and it produces essentially the same stability result if we equate $\lambda$ with the Jacobian $J$ at a given point. An important caveat, however, is that $\lambda$ is constant, whereas the Jacobian $J$ varies for a nonlinear equation, and hence the stability can potentially change.

# 数学物理学的高级课题 Advanced Topics in Mathematical Physics MATH3351/MATH6211

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$$\sigma_{y}=\sigma F-(T F) \sigma$$
where
$$F=u \sigma+v+w\left[T^{-1}(\sigma)\right]^{-1} \text {. }$$
One can check the by direct substitution of the operator $\sigma$ and by use of the equation for $\phi$.
Remark 2.33. Theorem $2.32$ is evidently valid for the spectral problem
$$\lambda \psi=u T \psi+v \psi+w T^{-1} \psi$$
with the only correction being that the last term for the transform $v[1]$ is absent. The equation goes to the “Riccati equation” analog for the function $\sigma$ :
$$\mu=u \sigma+v+w\left[T^{-1}(\sigma)\right]^{-1}$$

## MATH3351/MATH6211COURSE NOTES ：

$$\xi_{t}=[v / 2+(u+\beta I) T] \xi=Z \xi$$
which is solved by
$$\xi=\exp (Z t) \xi_{0} .$$
Plugging $\Phi$ into (2.135) yields the spectral problem for the difference shift operators:
$$\mu \Phi(x)=\xi^{-1}[u \xi \Phi(x+1)+v \xi \Phi+w \xi \Phi(x-1)]$$
Separating variables again, a class of particular solutions is built as
$$\Phi=\eta \exp (\Sigma x)$$
hence, we arrive at the matrix spectral problem for $\eta$ :
$$\mu \eta=\xi^{-1}[u \xi \eta \exp (\Sigma)+v \xi \eta+w \xi \eta \exp (-\Sigma)]$$
with the operator on the right-hand side and, therefore, spectral parameter $\mu$ parameterized by $t$. Finally, the matrix $\sigma$ is composed as
$$\sigma=\xi(t) \eta \exp (\Sigma) \eta^{-1} \xi^{-1}(t)$$

# 高级代数2：场扩展和伽罗瓦理论 Advanced Algebra 2: Field extensions and Galois Theory MATH3345/MATH6215

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Thus, assume that each $a_{i}$ is $n$-powerless. Observe that in this case it suffices to prove the theorem for $n_{1}=\cdots=n_{t}=n$ as clearly
$$\left(\mathbf{Q}\left(\sqrt[\pi t]{a_{1}}, \ldots, \sqrt[n_{t}]{a_{t}}\right) / \mathbf{Q}\right) \leq n_{1} \cdots n_{t}$$
and
$$\left(\mathbf{Q}\left(\sqrt[n]{a_{1}}, \ldots, \sqrt[n]{a_{t}}\right) / \mathbf{Q}\left(\sqrt[n]{a_{1}}, \ldots, \sqrt[n_{t}]{a_{t}}\right)\right) \leq\left(n / n_{1}\right) \cdots\left(n / n_{t}\right)$$
So
$$\left(\mathbf{Q}\left(\sqrt[n]{a_{1}}, \ldots, \sqrt[n]{a_{t}}\right) / \mathbf{Q}\right)=n^{t}$$
forces both inequalities to be equalities.

## MATH3345/MATH6215COURSE NOTES ：

In the above situation, let
$$\theta=\alpha_{1} Y_{1}+\cdots+\alpha_{n} Y_{n} \in \tilde{\mathbf{E}}$$
and let
$$F(Z)=\prod_{\tau^{\prime} \in S_{n}}\left(Z-\tau^{\prime}(\theta)\right) \in \tilde{\mathbf{E}}[Z] .$$
The polynomial $F(Z)$ is a symmetric function of its roots, so by its coefficients are functions of the elementary symmetric functions of its roots and hence of $Y_{1}, \ldots, Y_{n}$ and the coefficients of $f(X)$. Thus, $F(Z) \in$ $\tilde{\mathbf{F}}[Z]$. Now we may factor $F(Z)$ into a product of irreducibles in $\tilde{\mathbf{F}}[Z]$,
$$F(Z)=F_{1}(Z) \cdots F_{t}(Z)$$
One of these is divisible by $Z-\theta$ in $\tilde{\mathbf{E}}(Z)$. Renumbering if necessary we may assume it is $F_{1}(Z)$. Since $F(Z)$ is invariant under $S_{n}$, the action of $S_{n}$ permutes these factors.

# 高级分析2：勒贝斯格积分和希尔伯特空间 Advanced Analysis 2: Lebesgue Integration and Hilbert Spaces MATH3320/MATH6212

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Suppose that a function $f(x)$ is bounded on the interval $[A, B]$, where $A, B \in \mathbb{R}$ and $A<B$.
DEFINITION. The real number
$$I^{-}(f, A, B)=\sup {\Delta} s(f, \Delta),$$ where the supremum is taken over all dissections $\Delta$ of $[A, B]$, is called the lower integral of $f(x)$ over $[A, B]$. DEFINITION. The real number $$I^{+}(f, A, B)=\inf {\Delta} S(f, \Delta),$$
where the infimum is taken over all dissections $\Delta$ of $[A, B]$, is called the upper integral of $f(x)$ over $[A, B]$.

REMARK. Since $f(x)$ is bounded on $[A, B]$, it follows that $s(f, \Delta)$ and $S(f, \Delta)$ are bounded above and below. This guarantees the existence of $I^{-}(f, A, B)$ and $I^{+}(f, A, B)$.

## MATH3320/MATH6212COURSE NOTES ：

$$S(f, \Delta)-s(f, \Delta)<\frac{\epsilon}{c}$$ It is easy to see that $$S(c f, \Delta)=c S(f, \Delta) \quad \text { and } \quad s(c f, \Delta)=c s(f, \Delta)$$ Hence $$S(c f, \Delta)-s(c f, \Delta)<\epsilon .$$ It follows from Theorem $2 \mathrm{D}$ that $c f \in \mathcal{R}([A, B])$. Also, (13) clearly implies $I^{+}(c f, A, B)=c I^{+}(f, A, B)$. Suppose next that $c<0$. Since $f \in \mathcal{R}([A, B])$, it follows from Theorem $2 \mathrm{D}$ that for every $\epsilon>0$, there exists a dissection $\Delta$ of $[A, B]$ such that
$$S(f, \Delta)-s(f, \Delta)<-\frac{\epsilon}{c}$$
It is easy to see that
$$S(c f, \Delta)=c s(f, \Delta) \quad \text { and } \quad s(c f, \Delta)=c S(f, \Delta)$$
Hence
$$S(c f, \Delta)-s(c f, \Delta)<\epsilon .$$
It follows from Theorem $2 \mathrm{D}$ that $c f \in \mathcal{R}([A, B])$. Also, (14) clearly implies $I^{+}(c f, A, B)=c I^{-}(f, A, B)$.
(c) Note simply that
$$\int_{A}^{B} f(x) \mathrm{d} x \geq(B-A) \inf _{x \in[A, B]} f(x)$$
where the right hand side is the lower sum corresponding to the trivial dissection.
(d) Note that $g-f \in \mathcal{R}([A, B])$ in view of (a) and (b). We now apply (c) to the function $g-f$.
Next, we investigate the question of breaking up the interval $[A, B]$ of integration.