MATH3022 Scientific and Industrial Modelling/MATH3075 Financial Derivatives (Mainstream)/STAT1070/COMP1350 Introduction to Database Design and Management

Mon 19 Sept 3 pm

1. Fourier Series Convergence Problems:
(a) Obtain a Fourier expansion for f(x) = xexp(x) over 0 < x < 1 using the eigen-
functions arising out of the Sturm-Liouville problem
φ 00 (x) + λ 2 φ(x) = 0, with φ 0 (0) = 0, φ(1) = 0.
(b) Plot out the eigen-solution approximation using 4 terms over the extended do-
main −1 < x < 2. Comment on the usefulness/convergence of the result for
representing f(x) over 0 < x < 1, and indicate ‘why’ this result is to be expected.
(c) One can obtain an improved approximation by first extracting out an appropri-
ate linear function from f(x) and representing the remaining term as a Fourier
expansion. Check this out by writing f(x) = g(x) + (cx + d) and choosing (c,d)
appropriately. Plot out f(x) together with the above two representations using 4
terms in each case. Comment.
2. Complete the cylindrical rod cooling problem from lectures, see Fourier series lecture
S-L theory example:
An infinite cylindrical rod (radius a) is initially at temperature T = T 0 and its surface
temperature is reduced to T = 0. Determine the time required for the cylinder to cool.
To do this:
(a) Write down the defining equations and scale the problem to reduce it to the form:
1
r
(rT r ) r = T t with T(r,0) = 1, and T(1,t) = 0, T(0,t) finite.
(b) Proceed as in lectures to obtain a Bessel function expansion of the solution and
evaluate the coefficients so that the initial condition is satisfied.
(c) Plot out the solution as a function of r at t = 0 to see how many terms are
required for a good answer.
(d) Plot out the temperature at r = 0 and thus determine the cooling time in unscaled
terms as a function of radius etc.
3. An Explosion:
The spherically symmetric form of the diffusion equation for the dispersal of a material
in a medium is given by
c t = κ(c rr +
2
r
c r ),
where c is the concentration of the material. The associated 3D point source (or
fundamental) solution is given by
c(r,t) = Me −r
2 /(4κt) /[8(πκt) 3/2 ], where r =
q
x 2 + y 2 + z 2
where M is the mass of material released at the origin r = 0 at t = 0.
(a) To show that the above expression is the point solution:
i. Write down all the equations determining this solution and carefully argue
on dimensionality grounds that the structure of the solution is given by
c(r,t) =
MC(s)
(κt) 3/2
, where s = r 2 /(κt).
ii. Substitute this form into the diffusion equation to obtain the second order
ordinary differential equation for the function C(s) given by
4sC 00 [s]) + (6 + s)C 0 [s] + (3/2)C[s] = 0.
Determine the appropriate boundary and additional conditions that need to
be imposed on C[s]. Solve this equation with boundary and integral conditions
using Mathematica and thus show that the fundamental point solution is as
above.
(b) An explosion at a height H above the ground releases a mass M of toxic material
into the atmosphere. Subsequently the material diffuses away from the point
source. The atmosphere is assumed to be semi-infinite in extent and the ground
impermeable to the material.
Using the point source solution from part (a) determine an expression for con-
centration levels as a function of time at the point at ground level immediately
below the release point.
By introducing an appropriate time scale t 0 and concentration scale c 0 express
your result in the form c = c 0 c 0 (t 0 t 0 ) where c 0 and t 0 are dimensionless and all
parameters are removed from the description; this allows for presenting ‘all’ results
(for different parameter values) on the one graph. Sketch c 0 (t 0 ). Determine the
(scaled and then) unscaled maximum concentration level reached at ground level
and the time taken to reach this value, and comment on the dependence of these
results on H,κ.
4. A medication pill (here assumed to be spherical of radius a) is produced by compressing
together a small amount of active ingredient with a binding (and processing) powder.
The active ingredient diffuses through the pill ‘matrix’ into the stomach and then into
the blood stream. (After the release of the active ingredient the pill matrix breaks up.)
The release rate is medically important and may be adjusted by changing the amount
active ingredient and binder, and level of compression used when producing the pill.
A model of the diffusion process within the pill is given by:
∂c
∂t
= κ∇ 2 c in r < a with c(r,0) = c 0 , and c(a,t) = 0,
with
∇ 2 c(r,t) = c rr +
2
r
c r .
Here c(r,t) is the concentration of active ingredient in the pill. The concentration
outside the pill is assumed to be zero (a rapid removal rate in the gut).
(a) Scale the equations appropriately
c(r,t) = c 0 c 0 (r 0 ,t 0 )
etc.
(b) Solve for c 0 (r 0 ,t 0 ) and plot out solutions for various values of t 0 . To do this:
i. Obtain separated solutions of the boundary value problem. Identify the
Sturm-Liouville theory weighting function and use the orthogonality con-
dition to determine the Fourier solution.
ii. Check to see how many terms are needed to obtain a useful result.
iii. Plot out the solution for various times.
(c) How long does it take for ‘all’ the active ingredient to be released?
(d) Plot out the (scaled) release rate as a function of (scaled) time.
Pills are normally disc shaped. Why do you think this is so?
5. Chemical Release:
A chemical is released at x = 0 into a long thin channel containing a solid matrix
and diffuses along the channel. The channel is of thickness d, length L ? d and is
infinite width. The relevant equations determining concentration levels c(x,z,t) along
the channel are
c t = κ(c zz + c xx ), with c(x,z,0 − ) = 0,
and
Z
∞
−∞
dx 0
Z
d
0
c(x 0 ,z 0 ,t)dz 0 = Md,
where M is the total amount of chemical released per unit ares of channel ‘at’ t =
0,x = 0.
(a) Scale the equations and thus show that if
d
L
= δ ? 1 then the equations reduce
to a 1D diffusion model.
(b) For ‘small time’ one might expect conditions at the end of the channel to not
matter as for as the solution is concerned. Use dimensionality arguments to
determine a similarity solution form under such circumstances.
(c) Obtain this solution.
(d) For what time scale would you expect this solution to apply?
(e) By adding in an image source obtain an improved solution for the finite length
channel.
6. (a) Find the eigenvalues and eigenfunctions of
(x 2 φ 0 ) 0 + λx 2 φ = 0, (1)
with φ 0 (0) = φ(1) = 0.
Hint: Substitute φ(x) = Φ(x)/x, or use Mtica.
(b) Find the eigenvalues and eigenfunctions of (1) above with
i. φ(1) = 0, φ(3) = 0.
ii. φ 0 (0) = 0, φ(1) + φ 0 (1) = 0
7. Starting a Fire?
What thermal and chemical characteristics of a material determine its combustibility,
and what heating pattern is most likely to cause ignition? Ignition occurs if the tem-
perature exceeds the ignition temperature of the material, so the problem reduces to
one of determining the maximum temperature realised with prescribed heating.
A given amount of heat per unit area H (constant) is applied to a surface of a body
over a time interval ∆.
(a) Assuming the heating rate is uniform over the interval, use (7.29) (see notes), to
show that the maximum temperature is reached at the surface of the material at
the end of the heating interval. Determine this maximum value as a function of
∆,H, and the material properties.
(b)* Using Mathematica determine the maximum temperature reached as a function
of the index n if the rate of heating is a suitable multiple of t n for n = −1/2,1,2,3.
Compare the results with those obtained if the heating rate is uniform or if all
the heat is supplied instantaneously. Interpret your results.
(c)* If it is known that there is a reasonable spread of the heat supply over the time
interval, what temperature rise would you predict? Estimate the amount of heat
necessary to start the reaction as a function of the ignition temperature T ign ,
the material properties, and ∆. Comment on the material properties that are
desirable for combustibility and incombustibility.
(d) What do the above results tell us is the best way to start a fire in a combustible
material?
Note: The ∗’d questions are not for assessment.
8. Lake Pollution
Phosphate from agricultural land is flushed into the upper layers of a lake of depth h
at t = 0, and then diffuses downwards. No further phosphate is flushed into the lake.
The mass flux per unit area transferred downwards by turbulent diffusion is described
by m = −κ∂c(z,t)/∂z where c(z,t) is the concentration at depth z at time t; so the
diffusion equation
c t = κc zz
governs the process.
(a) Explain why, in the absence of deposition on the lake’s bottom the constraint
Z
h
0
c(z,t)dz = M
is reasonable, where M is the mass of phosphate compound per unit lake surface
area flushed into the lake at t = 0.
(b) For small time it is to be expected that the fundamental one dimensional source
solution will accurately describe dispersal. Write down this solution and indicate
the equations this solution satisfies.
(c) Over what time scale would you expect the similarity solution to accurately de-
scribe c(z,t)?
(d) By adding to the above solution a contribution due to a second source, obtain a
description that’s useful over a larger time range, and estimate the time scale for
which this solution is useful. Assume no deposition.
(e) ? Make further improvements and plot successive approximations.
(f) For large time what would you expect the concentration to be, assuming no depo-
sition?
(g) ?? Assuming deposition to the lake’s bottom occurs at a rate that’s proportional
to the concentration there, obtain an integral equation for the deposition rate,
and suggest a useful first approximation for early time.
Hint: Introduce an additional sink of unknown strength at the lake’s bottom,
and ensure there’s no additional flux introduced at z = 0.
9. Nuclear Fallout
This problem and the problem that follows trace some of the steps involved in attempt-
ing to model the fall-out from a nuclear accident. The primary objective is to determine
atmospheric concentration levels and fall-out levels at various locations. Usually the
amount of material released will be unknown, so it would be necessary to infer this
from measured concentration levels. A crude dispersal model which ignores fall-out is
examined here.
The transport of fine nuclear material expelled into the atmosphere is dominated by
air movement which may be thought of as having a steady velocity component (with
non-zero mean) with turbulent fluctuations (with zero mean) superimposed. Basically
the mean flow carries the nuclear material with it, while the fluctuations cause mix-
ing and thus dispersal of the material. It may be shown theoretically and displayed
experimentally that the rate and direction of dispersal of a material due to turbulent
fluctuations depends on the concentration gradient, so the situation is analogous to
heat dispersal with the mass flux per unit area given by m = −κ∇c where c is the
concentration of material and the dispersion coefficient, κ, will depend on the size of
the turbulent fluctuations; experimental data is available.
We’ll assume the mean velocity and the dispersion coefficient remain fixed, and we’ll
use a co-ordinate system moving with the mean flow. We’ll examine the situation in
which the accident releases a mass M of material at t = 0 at (x,y,z) = (0,0,0), with
0 < z < h, where h is the effective height of the atmosphere. After this release the
leak is plugged.
In the initial stages the dispersal process will be very complicated and dependent on
the details of the accident. However, after a time scale of order h 2 /κ, one might
expect the concentration profile in the vertical direction to settle down; for simplicity
let’s assume for this preliminary model that the concentration is independent of z, so
c(x,y,z,t) ≡ c(x,y,t); and that the fall-out is negligible.
Under the prescribed circumstances c(x,y,z) ≡ c(r,t) where r =
√ x 2
+ y 2 (why?), so
that it’s appropriate to use the cylindrical form of the Laplacian and the dispersal is
governed by
∂c
∂t
= κ(
∂ 2 c
∂r 2
+
1
r
∂c
∂r
).
(a) Use scaling arguments to show that the similarity solution form is given by
c(r,t) =
M
κht C(ξ)
where ξ = r 2 /κt, and determine the equation for C.
(b) Use Maple to obtain the similarity solution explicitly.
(c) Plot C(ξ) and comment on the solution behaviour.
(d) Plot c(r 0 ,t) for various distances r 0 measured from the origin in the moving frame
(ie. the effective source location). Comment.
(e) Plot the maximum (scaled) concentration levels expected at various distances r 0
from the effective source location, and determine the expected time for concen-
tration peaks. A typical figure for κ is 5 × 10 4 cm 2 /s and for mean wind speed,
15m/sec.
(f) Although from the scientist’s point of view the above plots are of most interest,
civil authorities would like to know about the changes in concentration levels at
fixed locations on the earth. Plot scaled concentration levels at various scaled
locations downstream from the accident as a function of scaled time.

=====================================================================================

MATH3075 Financial Derivatives (Mainstream)
11:59 p.m. on Sunday, 11 September 2022

1. [12 marks] Single-period multi-state model. Consider a single-period market
model M = (B, S) on a finite sample space Ω = {ω1, ω2, ω3}. We assume that the
money market account B equals B0 = 1 and B1 = 4 and the stock price S = (S0, S1)
satisfies S0 = 2.5 and S1 = (18, 10, 2). The real-world probability P is such that
P(ωi) = pi > 0 for i = 1, 2, 3.
(a) Find the class M of all martingale measures for the model M. Is the market
model M arbitrage-free? Is this market model complete?
(b) Find the replicating strategy (φ
0
0
, φ1
0
) for the contingent claim X = (5, 1, −3)
and compute the arbitrage price π0(X) at time 0 through replication.
(c) Compute the arbitrage price π0(X) using the risk-neutral valuation formula
with an arbitrary martingale measure Q from M.
(d) Show directly that the contingent claim Y = (Y (ω1), Y (ω2), Y (ω3)) = (10, 8, −2)
is not attainable, that is, no replicating strategy for Y exists in M.
(e) Find the range of arbitrage prices for Y using the class M of all martingale
measures for the model M.
(f) Suppose that you have sold the claim Y for the price of 3 units of cash. Show
that you may find a portfolio (x, φ) with the initial wealth x = 3 such that
V1(x, φ) > Y , that is, V1(x, φ)(ωi) > Y (ωi) for i = 1, 2, 3.
2. [8 marks] Static hedging with options. Consider a parametrised family of
European contingent claims with the payoff X(L) at time T given by the following
expression

X(L) = min

2|K − ST | + K − ST , L

where a real number K > 0 is fixed and L is an arbitrary real number such that
L ≥ 0.
(a) Sketch the profile of the payoff X(L) as a function of the stock price ST and
find a decomposition of X(L) in terms of terminal payoffs of standard call and
put options with expiration date T. Notice that the decomposition of the payoff
X(L) may depend on values of K and L.
(b) Assume that call and put options are traded at time 0 at finite prices. For
each value of L ≥ 0, find a representation of the arbitrage price π0(X(L)) of
the claim X(L) at time t = 0 in terms of prices of call and put options at time
0 using the decompositions from part (a).
(c) Consider a complete arbitrage-free market model M = (B, S) defined on some
finite state space Ω. Show that the arbitrage price of X(L) at time t = 0 is a
monotone function of the variable L ≥ 0 and find the limits limL→3K π0(X(L)),
limL→∞ π0(X(L)) and limL→0 π0(X(L)) using the representations from part (b).
(d) For any L > 0, examine the sign of an arbitrage price of the claim X(L) in any
(not necessarily complete) arbitrage-free market model M = (B, S) defined on
some finite state space Ω. Justify your answer.

=====================================================================================

STAT1070
11:59pm, Sunday 25 September.

Question 1. [12 marks]
Which university campus tends to have warmer maximum temperatures in September? To test this,
a random sample of 23 September days was taken from 2016–2020 and the maximum temperatures
were recorded at the Newcastle University and Gosford weather stations, the nearest stations to the
Callaghan and Central Coast campuses, respectively. The file campustemp.omv contains the data,
which includes the following variables:
• Year: the year of the recording
• Month: the month of the recording
• Day: the day of the recording
• Newcastle: the reported daily maximum temperature at the Newcaslte University weather
station
• Gosford: the reported daily maximum temperature at the Gosford weather station

(a) [1 mark] Are the September daily maximum temperature observations at the Newcastle Uni-
versity and Gosford stations paired or independent? Write a sentence justifying your choice.

(b) [6 marks] Is there evidence that the average daily maximum temperature in September differs
between the Newcastle University and Gosford weather stations? Conduct the appropriate test
in jamovi and include all relevant output. Be sure to define any parameters you use, state the
null and alternative hypotheses, observed test statistic, null distribution, p-value, decision and
provide an appropriate conclusion in plain language.
(c) [2 marks] Report the 95% confidence interval for the average difference in the daily maximum
temperature in September between the Newcastle University and Gosford weather stations. Write
a sentence interpreting this interval in plain language.
(d) [1 mark] Does your confidence interval from part (c) support the decision you made in part
(b)?

(e) [2 marks] What are the assumptions of your analyses in parts (b) and (c)? Are these assump-
tions met? Justify why or why not for each assumption, with appropriate references to jamovi

output where needed.

Question 2. [14 marks]

Over time, the way that music has been consumed has changed, shifting from promotion via phys-
ical products, radio play, and music videos to digital downloads and streaming through audio and

video platforms such as Spotify, Apple Music, and YouTube. To account for changing methods of
consumption, various national charts have adapted their methodologies accordingly. Is there a shift
in the typical lengths of songs that become popular which is associated with the changing ways of
music consumption and chart accounting? To test this, a random sample of 50 songs that reached
the Top 40 of the U.S. Billboard Hot 100 was taken from each of the 1980s, 1990s, 2000s, 2010s, and
2020s, with the data recorded in the file billboard.omv.
The file includes the following variables:
• Artist: the name of the artist(s)
• Song: the song title
• Peak: the peak position on the Hot 100 chart
• Decade: the decade the song reached the top 40
• Length: the length of the song, in seconds

(a) [2 marks] Produce a side-by-side boxplot and descriptive statistics to explore the relationship
between decade and song length of U.S. Top 40 hits. Describe the relationship between the
length of the songs and the decade in which they charted on the Hot 100.
(b) [6 marks] Is there evidence of a difference in average length of songs that charted in the U.S.
Top 40 among the five decades assessed? Be sure to state the null and alternative hypotheses,
test statistic, null distribution, p-value, decision and an appropriate conclusion in plain language.
(c) [3 marks] If appropriate, perform post-hoc tests to determine which decades have significantly
different average song lengths. If post-hoc tests are not appropriate, explain the purpose of a
post-hoc test and why it’s not appropriate in this example.
(d) [3 marks] What are the assumptions of the analysis performed in part (b)? State whether
each assumption is reasonable with reference to appropriate jamovi output.

Question 3. [20 marks]
The file downloads.omv contains observations of the amount of time someone spent online and
the amount of memory they downloaded for 40 randomly sampled clients. The variables included in
the dataset are:

• Time: the amount of time the client was online (in minutes)
• Memory: the amount of memory downloaded while online, in megabytes

(a) [2 marks] Generate a scatterplot with the amount of memory downloaded on the y-axis and
the time online on the x-axis and add the fitted regression line. Briefly describe the relationship.

(b) [3 marks] Write down the equation for the estimated regression line and provide an interpre-
tation of the intercept and the slope coefficient.

(c) [1 mark] Predict the amount of memory downloaded for a client who spends 20 minutes online.
(d) [6 marks] Is there a statistically significant linear relationship between the amount of memory
downloaded and the time someone spends online? Be sure to state the null and alternative
hypotheses, test statistic, null distribution, p-value, decision and an appropriate conclusion in
plain language.
(e) [4 marks] State the assumptions necessary for your regression analysis in part (d) to be
appropriate. State whether each of them is satisfied with a brief justification. This justification
may refer to appropriate output from jamovi.
(f) [2 marks] Provide a 95% confidence interval for the slope of the population regression line
of the amount of memory downloaded on the amount of time someone spends online. Write an
interpretation of this interval.
(g) [2 marks] Write down the R2 value for this regression and give an interpretation.

==================================================================================

COMP1350
11.55 pm Sunday 04 September 2022

Case Background
Hiraeth Cruises has been in the cruise business for the last 30 years. They started storing data in
a file-based system and then transitioned into using spreadsheets. As years progressed, their
business has grown exponentially leading to an increase in the number of cruises and volume of
data. You have been recently employed a database model to replace the current spreadsheets.
You have been provided with the following business rules about Hiraeth Cruises. This is only a
section of their business rules.
Vessels: Every vessel is uniquely identified using a primary identifier. Other details of a vessel
include the name, and the year it was purchased. Every vessel is of a particular model. Every
model is identified with a unique identifier. The name of the model and the passenger capacity
for the model are recorded. The vessels are serviced in service docks. Every service dock is
identified using a primary identifier. The name of the dock is also recorded. A vessel could get
serviced in multiple docks. Every time the vessel is serviced, the service date, and multiple
comments about the service are stored. There are three types of vessels: small, medium, and large
vessels.
Cruises: Every cruise is uniquely identified using a primary identifier. Other details of a cruise
include the name, and the number of days the cruise goes for. There are two types of cruises: short
and long cruises. A vessel gets booked by the cruise for a few months which are also recorded.
The short cruises use small vessels, whereas the long cruises use either a medium or a large
vessel.
The cruises are created along a particular route. Every route is identified using an identifier. The
description of the route is also stored. A route will have a source location, a destination location
and multiple stopover locations. Each location is identified by a location code. The name of the
location is also stored.
Tours: Every cruise offers a unique set of tours for their customers. A tour code is used to
identify a tour within every cruise. Other details of the tour such as the name, cost, and type are
stored. A tour could be made up of other tours (a package). A tour could be a part of multiple
packages. A tour will belong to a particular location. A location could have multiple tours.
Staffing: Every Hiraeth staff member is provided with a unique staff number. The company also
needs to keep track of other details about their staff members like their name, position, and
their salary. There are two types of staff that need to be tracked in the system: crew staff and
tour staff. For crew staff, their qualifications need to be recorded. For tour staff, their tour
preferences need to be recorded. There are three types of tour staff – drivers, our guides, and
assistants. The license number is recorded for the driver and the tour certification number is
recorded for the tour guide. In certain instances, the drivers will need to be tour guides as well.
Tour staff work for a particular location.
Scheduling: A schedule gets created when the cruise is ready to handle bookings. The start date
and the max capacity that can be booked are recorded. Every schedule has a detailed roster of the
staff involved in the cruise including the crew and the tour staff. The start and end time for every
staff will be stored in the roster.

Task Description
Task 1- EER Diagram (60 marks)
Based on the business rules, you are expected to construct an Enhanced-ER (EER) diagram. The
EER diagram should include entities, attributes, and identifiers. You are also expected to show
the relationships among entities using cardinality and constraints. You may choose to add
attributes on the relationships (if there are any) or create an associative entity, when necessary.
Your diagram should also specify the complete (total) and disjoint (mutually exclusive)
constraints on the EER.
Task 2- Logical Transformation (40 marks)
Based on your EER, perform a logical transformation. Please use 8a for your step 8 to keep the
process simple. Please note, if there are errors in the EER diagram, this will impact your marks in
the transformation. However, the correctness of the process will be taken into account.