Changing the Undergraduate Experience

Kevin Golden

University of the West of England

Department of Mathematics and Statistics, Bristol Institute of Technology, UWE

Lets start at the endA: Software developer working on efficient search engine algorithms.

D: Credit risk analyst for Barclaycard

E: GSK fast track management training scheme

F: Secondary school teacher

G: PhD in Metereology

H: UK Transplant centre working on donor pairing scheme

I: Government Statistician working for Office of National Statistics

C: MOD Statistician working in epidemiology

B: Airbus fast track management training scheme

Moving back a bit

Final Year Projects

DefinitionsFIXED POINT: The point x* say is said to be a fixed point of if f(x*)=0. For the pendula systems, this is a point where that if the pendulum is released from rest at x*

it will remain there for all time.

STABLE: A fixed point is stable if all solutions starting sufficiently close to the fixed point remain close to the fixed point for all time.

UNSTABLE: A fixed point is unstable if there exists solutions starting arbitrarily close to the fixed point which move away from the fixed point as t increases.

BIFURCATION DIAGRAM: A graphical representation of the set of all fixed points in a system for a bifurcation parameter and the dependent variable(s).

Nonlinear Systems of the Simple PendulumSimon Warren University of the West of England

Introduction and AimsA simple pendulum consists of a mass m, suspended from a fixed point by a rod of length L, and moving freely under the force due of gravity. If instead the mass is now attached to a wire rather than a fixed point, where the wire is horizontal and fixed at each end - and has a torsional constant β, the torsional pendulum is obtained. Attaching a second, identical pendulum to this wire, the torsion coupled pendula is obtained.

I am going to model and understand the motion of these pendula systems, determining where fixed points occur and their corresponding stability. Where torsion is a parameter of the system, the effects of the torsion on the system will be investigated.

Methods

• Derivation of equations of motion using resolved forces.• Location of fixed points by stability definition.• Use of potential energy curves to classify the stability of fixed points.• Symmetry of initial conditions to simplify solutions.• Phase portraits and time plots to visualize and understand solutions.• Bifurcation diagrams to represent the set of fixed points and stability.

All data and graphs provided on this poster are created using the parameters m=1, L=1 and g=9.81.

The Simple PendulumThe equation of motion for the simple pendulum is given by eqn(1.1), and can be linearised to the form of eqn(1.2).

0 (1.1)

sin 0 (1.2)

gx x eqnLg

x x eqnL

The potential energy curve above relates directly to the nonlinear phase portrait. As seen the peaks of the curve match the fixed points in the phase portrait. Furthermore we can see that the fixed points of the nonlinear system occurring at angles x=±nπ, where n

= …-2,-1,0,1,2 … And the stability of these fixed points alternate where n=-2,0,2,.. are stable centres, and n= -3,-1,1,3.. are unstable nodes.

The Torsional Pendulum –A simple pendulum attached to a wire with torsion

The equation of motion for the torsional pendulum is given by eqn(2.1). Where β>0 is the torsional constant of the wire.

2

2sin 0 (2.1)g

x x x eqnL mL

sin , (2.2)

2where > 0

x x eqn

mgL

The phase portrait (top right), shows how various solutions to initial conditions change over time for a chosen γ (γ=0.15). The potential energy plot (bottom right) indicates the stability of the fixed points. As can been seen, the fixed points alternate in stability either side of the stable fixed point x=0. In the phase portrait, solutions sufficiently close to stable fixed points form stable centres, where as unstable solutions form saddle nodes. The potential well solutions can also be seen on the phase portrait.

The Torsion Coupled Pendula – Two simple pendula coupled by a wire with torsion

1 1 1 22

2 2 2 12

sin 2 0

sin 2 0 (3.1)

gx x x x

L mLg

x x x x eqnL mL

The equations of motion for the torsion coupled pendula is given by the system of differential equations as given in eqn(3.1). Where β>0 is the torsional constant of the wire.

The fixed points of this system obtained by the solutions to eqn(3.1), (where γ is as defined in eqn(2.2)). The bifurcation diagram of the set of all fixed points is now three dimensional in the two space parameters x1 and x2 and the bifurcation parameter γ. However, since a 3-D plot is hard to visualise, two separate bifurcation diagrams in x1 and γ, and x2 and γ. Displayed is the bifurcation diagram in x1.

1 1 1 1 1

2 2 2 2 2

2 3'( , ) sin sin 2 2sin 0, (3.2)

2

2 3'( , ) sin sin 2 2sin 0. where >0

2

f x x x x x eqn

g x x x x x

The motion of the torsion coupled pendulum were found to follow three trends, dependent on initial conditions.

1 2 1 21) (0) (0), (0) (0) 0x x x x

Solutions were such that x1(t)=x2(t) for all time. The time plots bellow show how the solutions do not differ over time. Where above is x1 against t and bellow x2 against t.

1 2 1 22) (0) (0), (0) (0) 0x x x x 1 2 1 23) (0) (0), (0) (0) 0x x x x

With initial conditions of this form, solutions were obtained which appeared quasiperiodic (bellow left), and chaotic (bellow right), however it is not true to say that all initial conditions of this form will have these solutions.

( )x f x

Potential Energy CurvesIn a system which is energy conservative, which our pendulum systems are, potential energy curves and surfaces indicate the stability of the fixed points of the system. Where, on the curve, the turning points are the locations of the fixed points. A maximum turning point indicates an unstable point, where as a minimum indicates a stable fixed point.

Imagine a smooth ‘mountain’, as pictured left. If a ball is balanced at the top of the peak, the ball will remain there for all time. However, if the ball is perturbed slightly, the ball will move away from this point – I.e. An unstable fixed point. Now, imagine a ball placed at the very bottom of a hill, the ball will remain there for all time. When perturbed slightly from this position, the ball will remain close to the fixed point – I.e. A stable fixed point

Potential wells: Furthermore, the potential curves can form potential wells. At these wells, solutions can be obtained such that if a ball is released from within a well, the ball will not escape the well, and so will oscillate only about the fixed points contained in the well.

Potential wells

PHASE PORTRAIT: A graphical representation of a set of solutions in the phase plane. We will generally be plotting the solutions of the angle x against its angular acceleration (unless otherwise stated) over a sufficient time period.

STABLE CENTRE: A stable fixed point for which solutions nearby do not converge to the fixed point. On a phase portrait they appear as concentric closed curves.

SADDLE POINT: A saddle point is an unstable fixed point. The majority of solutions move away from the fixed point.

The phase portrait shown left represents the set of solutions for the linear pendulum, and the phase portrait bellow for the exact nonlinear pendulum. Comparing the phase portraits we see that the linear approximation to the nonlinear equation is invalid over large angles.

The fixed points of the system are obtained by the solutions to eqn(2.2). The collection of fixed points in terms of γ are displayed in the bifurcation diagram bellow. As γ is decreased, the number of fixed points increase. The first few bifurcation points are shown in the opposite table.

x

0.058013

0.09139

0.2175

Bifurcation point

γ=

No of Fixed Points beyond γ

x2x1

From the bifurcation diagrams, we see that as the parameter γ is reduced, the number of fixed points increase.

Solutions were such that x1(t)=-x2(t) for all time. The time plots bellow show how the solutions are exactly out of phase over time. Where above is x1 against t and bellow x2 against t.

Quantitative Methods Project

The Statistical Development of Fixed Odds Betting Rules in Association Football

Literature• The gambling industry in Great Britain had an estimated turnover of £91.5 billion in 2006. [1]

• Fixed odds are offered on the home win, draw and away win of football matches. These odds are determined subjectively and do not change in response to betting volumes or new information. [2, 3]

• Profitable betting strategies can be developed if there are exploitable inefficiencies in the market. [2, 3, 4].

• Pankoff (1968) defines an inefficiency as occurring when “the market has failed to capture the information used by the superior analysts”. [5]

• Past research suggests avoid betting on American football games with heavy favourites. [6]

• Past research suggests that bettors should avoid the heavy underdogs. [6]

Within Sample

The most profitable betting strategy within sample was for betting on the away team with the binary regression approach when using the complementary log-log link function. Figure 1 shows the range of maximum profits available for each betting rule where the maximum return was £19.43 (a 67% profit return). This optimum was found when bets were made on predicted probabilities of outcomes that lie in the region,

0.44696 < Pi < 0.71461

The statistical model developed and the associated equation for calculating the probabilities of outcomes can be seen to the right. Using R and M also gave profits for the away win. Current literature only considers use of R and M, and not Pi.

Statistical Modelling• Need new, relevant predictor variables to develop profitable betting strategies.

• Newly considered predictor variables include the amount of time teams were winning and winning by more than two goals in their previous three games. This is a readily sourced combined measure of attacking and

defensive dominance. These predictors have not been considered in any previous literature.

• Predict bookmaker’s odds of matches using multiple linear regression.

• Difference between predicted odds and actual odds gives a measure of mismatch M.

• Predict probability (Pi) of match outcomes using logistic regression models.

• Ratio of predicted match probabilities and bookmaker’s odds gives a measure of mismatch R.

• Test out developed models with imaginary £1 bets.References[1] Gambling Commission, 2007. Annual Report and Accounts, 2006/07 [online]. Available

from:http://www.gamblingcommission.gov.uk/UploadDocs/publications/document/Annual%20Report%202006-07.pdf [Accessed 10 March 2008][2] Dixon, M. J., and Pope, P. F., 2004. The Value of Statistical Forecasts in the UK Association Football Betting Market. International Journal of Forecasting 20, pg. 697-711 [3] Goddard, J., and Asimakopoulos, I., 2004. Modelling football match results and the efficiency of fixed-odds

betting.[4] Dixon, M. J., and Coles, S. G., 1997. Modelling Association Football Scores and Inefficiencies in the Football Betting Market. Applied Statistics, 46, No. 2, pg. 265-280.[5] Pankoff, L. D., 1968. Market Efficiency and Football Betting. The Journal of Business, Vol. 41, No. 2, pg. 203- 214.[6] Woodland, L. M., 1994. Market Efficiency and the Favorite-Longshot Bias: The Baseball Betting Market. The Journal of Finance, Vol. XLIX, No. 1, pg. 269-279.

Ian Milliner Project Supervisor: Paul White

Abstract Rules

Betting on the away team is a risky strategy. Therefore opt for a low risk bet by betting when,

Mlower < M < Mupper

or if

Pilower < Pi < Piupper

or if

Rlower < R < Rupper

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191

Betting Rule

Ret

urn

Pro

fit

(£)

Going Live and Results

• The Complementary log-log model produced a profit of £3.14 from 5 bets made (a 63% profit return).

• The model had an 80% success rate at selecting the correct outcome on the first testing of the betting strategy.

• When tested a second time, the rule had a 100% success rate at selecting correct outcomes of games.

The results of this project are interesting since the most profitable betting strategy represents betting on the probabilities of outcomes where there is an upper limit to the betting region. This is contrary to all of the previous literature reviewed as this approach does not involve betting on those events with high probabilities, i.e. not betting on the favourite, nor the underdog, nor the maximum mismatch (so-called value bets). Instead bets are made on “middle of the road” high risk events of the away win. Optimised rules in the literature do not use an upper bound. However, using an upper bound has led to the development of more profitable betting rules. Results from this have been replicated too.

Figure 1: Attainable profits for betting rules when using Away Binary model to be away

RFO’A ‘0.0164871*RFO’ H‘0.0157026*-1.18966- )Cloglog(

))RFO’A ‘0.0164871* RFO’ H‘0.0157026* - 1.18966 - exp( - exp( - 1 ˆ

Current literature only considers a lower bound

Effect of Fluid Inertia on the Linear Viscoelastic Properties of Polymer Liquids

Faculty of Computing, Engineering and Mathematical Sciences

Sarah LeatAcademic Year 2007/2008

OBJECTIVESTo investigate the effect of fluid inertia on the complex viscosity function

SIMULATION AND CORRECTION ANALYSISComplex viscosity , is a complex number and the sum of the dynamic viscosity and the dynamic rigidity (The real and imaginary parts respectively)

Since in the simulations both were affected when fluid inertia was not taken into account and corrected for, it can be deduced that fluid inertia affects the complex viscosity function, in particular, the dynamic rigidity was most affected.

METHODS Mathematically described the effect of fluid inertia Simulated a controlled stress rheometer experiment in Maple to visualise the effect of fluid inertia on the dynamic rigidity and dynamic viscosity

CONCLUSIONSFluid inertia does have an affect on the complex viscosity function and if ignored, results can be misleading and wrong conclusions drawn.

TERMINOLOGYFluid Inertia - the resistance to the change in motion of a fluidLinear Viscoelastic Properties - linear relation between the stress and the applied rate-of-strainPolymer Liquids - materials exhibiting both viscous flow and elastic response properties when undergoing deformation i.e. are viscoelastic

CORRECTIONS FOR FLUID INERTIAThe most significant improvement after correcting for the effect of fluid inertia was seen on the dynamic rigidity curves.

Exact Curve Neglecting Fluid Inertia 1st Order Correction 2nd Order Correction 3rd Order Correction 4th Order Correction

Dyn

amic

Rig

idity

Frequency

/'' iG

VISCOELASTIC MATERIAL EXAMPLEA commonly found viscoelastic material is silly putty. This material will bounce when thrown gently, break when given a sharp blow and when left for a period of time, will form a puddle.

SIMULATION RESULTSNEWTONIAN LIQUIDThe most extreme example of the effect of ignoring fluid inertia occurred when simulating a Newtonian liquid.

When the complex viscosity is written as,

And the relaxation time, λ, is set to zero, the complex viscosity reduces to,

This means that for a Newtonian liquid the dynamic rigidity should be zero and the dynamic viscosity should be constant and equal the complex viscosity.

As the Maple output shows, the effect of ignoring fluid inertia effects greatly misrepresent the data.

Frequency

Frequency

Dynamic Viscosity

Dynamic Rigidity

i 1/

0

'

'G

Mathematics, Statistics and Operational Research at UWE

Between 35 and 40 students in each year

Case studies or project work are core in each year

Option of work-based learning education project in final year

Option to spend 3rd year placement in industry or commerce.

Graduate development programme delivered in each year

Group work in years 1 and 2 and an individual project in year 3.

Report writing, poster and oral presentations and project management.

1st year Problem Solving Modules, total of 40 credits, involving case studies and presentations

CORE

Core module in final year is an individual 20 credit project module. Students then choose taught modules from option list

Demand led programmes

Leitch report (2006) recommends the development of future HE programmes are demand led.

Implications are clear forPostgraduate programmes

CPD activity

Vocational UG programmes

What are the implications for

Non-vocational UG programmes ?

What do students want ?

Source: UWE Student Employability Survey 2005 – sample size 575 students

96% of UWE students think the University has a responsibility to help them become more employable

More employer involvement in course 60%

More placement opportunities 56%

More emphasis on skill development 47.5%

Increased career support 44.7%

HOW ?HOW ?

Advertising more job opportunities 41.1%

What do employers want ?

Source: UWE Employability Project – sample size 22 employers

Top 10 skills and AttributesTop 10 skills and Attributes

Team Working (S)

Motivation (A)

Initiative (A)

Problem Solving (S)

Independent Working (S)

Confidence (A)

Resilience (A)

Interpersonal Skills (S)

Emotional Intelligence (A)

Open Minded (A)

University of the West of EnglandGraduate Development Programme

Level 1Settling in

Development of learning skillsUnderstanding the university environment

Level 2Reconnection

Advanced academic skillsEmployability

Personal development

Level 3Reconnection

Academic achievementFinding employment

Lifelong learning

Retention

Engagement

Motivation

Performance

Satisfaction

Employment

The Mathematics GDP programme

Level 2

Reflection,

Expectations,

Employability,

Employment opportunities, Placements,

Final year projects

8 sessions

Level 1

Settling in, Expectations, Coursework, Group work, Research skills, Presentations, Feedback, Placements, Examinations.

12 sessions

Level 3

To be decided

Reflection

Expectations

Finding employment

CPD

Projects

4 sessions

Placement Years (Impact on graduate employment)

80% of employers recruited placement students with the primary aim of attracting them back to permanent jobs.

40% of annual graduate intake from these employers consisted of former placement students.

Source: Careers service, The University of Manchester and UMIST, (2004), [ASET website] http://www.asetonline.org

Placement Years (impact on academic performance)

Rawlings, P., White, P., and Stephens, R. (2006). Practice-Based Learning in Information Systems: The Advantage for Students, Journal of Information System Education, vol 16 (No 4): 449-458 [UWE]

Gomez, S., Lush, D and Clements, M. (2004). Work Placements Enhance the Academic Performance of Bioscience Undergraduates, Journal of Vocational Education and Training, Vol 56 (No 3): 373-385. [UWE]

Wallace, R.G. (2002). A Sandwich Year Can Seriously Damage Your Chances of Obtaining a Poor Degree”, ASET European Conference: Integrating Work & Learning, Cambridge, UK, Sept (2002). [Nottingham Trent]

Rawlings et al [14] demonstrate, in the context of information systems students, that the effect of a placement year could be shown to have significant positive impact on the final year performance of students from a wide range of academic abilities as measured by their second year performance, not just the most able.

Some Remarks

• At UWE, we have chosen integrate skills development into the programme

• Graduate Development Programme has created space to focus on skills development and employability

• Placement year seen as increasing in importance.

• Education project provides work-based learning opportunity.

• Skills development and employability viewed as core not optional