## 2000-level modules

#### MT2501  Linear Mathematics

Credits 15.0 Both 2019/20 12.00 noon Mon (odd weeks), Wed and Fri [Semester 1]; 11.00 am on Mon (even weeks), Tue and Thu [Semester 2] This module extends the knowledge and skills that students have gained concerning matrices and systems of linear equations. It introduces the basic theory of vector spaces, linear independence, linear transformations and diagonalization. These concepts are used throughout the mathematical sciences and physics. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT1002, or A at Advanced Higher Mathematics, or A at A-level Further Mathematics, or A at both A-level Mathematics and A-level Physics MT2001 2.5-hours lectures (x 10 weeks), 1 tutorial (x 4 weeks), 1 examples class (x 6 week) 2-hour Written Examination = 70%, Coursework (including class test) = 30% Prof N Ruskuc (S1), Dr A Wilmot-Smith (S2) Prof N Ruskuc (S1), Dr A Wilmot-Smith (S2)

#### Continuous assessment

• Class test (50 minute): 15%
• Coursework problem sets: 15%

#### Syllabus

• Matrices and determinants: basic revision of matrices & relevant fields (especially complex numbers); revision of e.r.o.’s; system of linear equations; determinants and their basic properties; matrix inverses; solutions of systems of linear equations.
• Vector spaces: Definition of vector spaces; examples of vector spaces (with emphasis on geometrical intuition); basic properties of vector spaces; subspaces.
• Linear independence and bases: spanning sets; linear independence, bases, dimension.
• Linear transformations: definition of linear transformation and examples (including trace), the matrix of a linear transformation; rank and nullity (including proof of Rank-Nullity Theorem); the rank of a matrix and reduced echelon form; rank and the matrix of a linear transformation.
• Eigenvalues, eigenvectors and diagonalization: eigenvalues and eigenvectors; change of basis; powers of matrices, symmetric matrices and quadratic forms.

• T.S. Blyth & E.F. Robertson, Basic Linear Algebra, Springer, 2002.
• R.B.J.T. Allenby, Linear Algebra, Edward Arnold, 1995.
• Richard Kaye & Robert Wilson, Linear Algebra, OUP, 1998.

#### MT2502  Analysis

Credits 15.0 1 2019/20 11.00 am Mon (even weeks), Tue and Thu The main purpose of this module is to introduce the key concepts of real analysis: limit, continuity and differentiation. Emphasis will be placed on the rigourous development of the material, giving precise definitions of the concepts involved and exploring the proofs of important theorems. This module forms the prerequisite for all later modules in mathematical analysis. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT1002 or A at Advanced Higher Mathematics or A at A-level Further Mathematics MT2002 2.5 hours lectures (x 10 weeks), 1-hour tutorial (x 5 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework (including 1 class test) = 30% Dr J M Fraser Dr J M Fraser

#### Continuous assessment

• Class tests (50 minute): 15%
• Fortnightly assessed tutorial questions: 5 x 3% = 15%

#### Syllabus

• The rationals and the reals: maximum & minimum, supremum & infimum, completeness.
• Sequences, series and convergence: the Bolzano-Weierstrass Theorem, tests for convergence - the ratio test, the root test, the comparison test, Cauchy sequences.
• Continuous functions: algebraic properties of continuous functions, the Intermediate Value Theorem.
• Differentiable functions: the chain rule, Rolle’s Theorem, the Mean Value Theorem, Taylor polynomials.

These topics will be introduced from a rigorous point of view, giving precise definitions, applying an ε-δ approach and giving examples.

• John M. Howie, Real Analysis, Springer, 2001, Chapters 1-4.
• Robert G. Bartle & Donald R. Sherbert, Introduction to Real Analysis, Wiley, 1992, Chapters 2-6.
• Kenneth Ross, Elementary Analysis, Spring, 1980, some parts of Chapters 1-6.

#### MT2503  Multivariate Calculus

Credits 15.0 1 2019/20 12 noon Mon (even weeks), Tue and Thu This module extends the basic calculus in a single variable to the setting of real functions of several variables. It introduces techniques and concepts that are used throughout the mathematical sciences and physics: partial derivatives, double and triple integrals, surface sketching, cylindrical and spherical coordinates. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT1002, or A at Advanced Higher Mathematics, or A at A-level Further Mathematics, or A at both A-level Mathematics and A-level Physics, or Co-requisite MT1010 MT2001 23 hours of lectures, 1-hour tutorial (x 4 weeks), 1-hour examples class (x 4 weeks) 2-hour Written Examination = 70%, Coursework = 30% (including 1 class test) Dr A P Naughton Prof A W Hood, Dr A P Naughton

#### Continuous assessment

• 50-minute class test: 15%
• Projects involving computer-based work: 15%

#### Syllabus

• Revision of basic differentiation rules: product rule, quotient rule, chain rule.  Hyperbolic functions & inverse hyperbolic function: graphs, derivatives, integrals & identities.
• Power series, including Taylor series about an arbitrary point.  Limits, continuity & differentiability of functions on one variable (definitions).  l’Hopital’s Rule.
• Revision of vectors and dot product.  Functions of several variables, representation as surfaces, surface sketching, and limits of functions of several variables, continuity and differentiability for functions of two variables.
• Partial derivatives, chain rule for functions of n-variables.
• Implicit differentiation and contours, higher order partial derivatives, derivatives in n-dimensions, tangent planes
• Taylor series for functions of two variables. Maxima and minima.
• Directional derivative and gradient. Lagrange multipliers.
• Revision of integration for functions of one-variable. Double integrals.  Spherical and cylindrical coordinates. Triple integrals.

• Earl W. Swokowski, Michael Olinick & Dennis Pence, Calculus, 6th ed., PWS Pub. Co., 1994.
• Erwin Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, 2011.
• Robert Adams & Christopher Essex, Calculus, 8th ed, Pearson 2013.

#### MT2504  Combinatorics and Probability

Credits 15.0 1 2019/20 4pm Mon (odd weeks), Thu and Fri This module provides an introduction to the study of combinatorics and finite sets and also the study of probability. It will describe the links between these two areas of study. It provides a foundation both for further study of combinatorics within pure mathematics and for the various statistics modules that are available. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT1002 or A at Advanced Higher Mathematics or A at A-level Further Mathematics, or Co-requisite MT1010 MT2004 or MT2005 2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 4 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework = 30% Prof C M Roney-Dougal Prof C M Roney-Dougal, Dr H Worthington

#### Continuous assessment

• Computer project: 15%
• Fortnightly assessed tutorial questions: 5 x 3% = 15%

#### Syllabus

• Counting & elementary probability: definition of sets, unions of disjoint sets, pigeonhole principle, notation needed for probabilities (e.g., events, complement); axioms of probability and concept of probability using counting argument; Inclusion-Exclusion.
• Basic rules of probability (building on elementary counting), conditional probability, multiplication rule, Bayes Theorem, independence.
• Ordered pairs, double-counting, size of Cartesian products of sets, choosing with repetition; functions, permutations.
• Recursion and generating functions: Binomial numbers, recursively and via generating functions; Fionacci numbers including some recursive formulae; Catalan numbers, including some recursive formulae; more on generating functions; counting partitions of a set.
• Random variables and distributions: definition of a discrete random variable (r.v.), probability mass functions, Bernoulli distribution, Binomial distribution, Poisson distribution, geometric distribution (including lack of memory property).
• Continuous r.v.s, probability density functions, uniform distribution; exponential distribution, normal distribution; cumulative distribution function (c.d.f., discrete & continuous cases), inverse c.d.f.
• Expectation; variance, introduction to probability generating functions; moment generating functions.
• Bivariate distributions: discrete/continuous distributions, joint, marginal and conditional probability mass/density functions; expectation, covariance and correlation, independence.

• Norman L. Biggs, Discrete Mathematics, 2nd ed., OUP, 2002.
• Ian Anderson, A First Course in Discrete Mathematics, Springer, 2001.
• John A. Rice, Mathematical Statistics and Data Analysis, Belmont, CA: Brooks/Cole CENGAGE, 2007.
• Richard D. De Veaux, Paul F. Velleman & David E. Bock, Stats: Data and Models, Pearson/Addison Wesley, 2005.

#### MT2505  Abstract Algebra

Credits 15.0 2 2019/20 11.00 am Mon (odd weeks), Wed and Fri This main purpose of this module is to introduce the key concepts of modern abstract algebra: groups, rings and fields. Emphasis will be placed on the rigourous development of the material and the proofs of important theorems in the foundations of group theory. This module forms the prerequisite for later modules in algebra. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT1002 or A at Advanced Higher Mathematics or A at A-level Further Mathematics MT2002 2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 5 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework = 30% Dr M R Quick Dr M R Quick

#### Continuous assessment

• Computer project: 15%
• Fortnightly assessed tutorial questions (4 weeks): 15%

#### Syllabus

• Preliminaries and prerequisites; equivalence relations
• The definitions and familiar examples of rings and fields
• The definition of a group, Cayley tables, elementary properties of groups
• Examples of groups: modular arithmetic including the Euclidean algorithm; permutation groups and symmetries
• The order of an element, subgroupscyclic groups, alternating groups, cosets and Lagrange’s Theorem
• Homomorphisms and isomorphisms, normal subgroups, ideals, quotient groups and rings, the First Isomorphism Theorem

• R.B.J.T. Allenby, Rings, Fields and Groups, 2nd ed., Edward Arnold, 1991.
• T.S. Blyth & E.F. Robertson, Essential Student Algebra, Vol.3: Abstract Algebra, Chapman & Hall, 1986.
• T.S. Blyth & E.F. Robertson, Algebra Through Practice, Book 3: Groups, Rings and Fields, CUP, 1984.
• D.A.R. Wallace, Groups, Rings and Fields, Springer, 1998.

#### MT2506  Vector Calculus

Credits 15.0 2 2019/20 9.00 am Mon (even weeks), Tue and Thu This module introduces students to some of the fundamental techniques that are used throughout the mathematical modelling of problems arising in the physical world such as grad, div and curl as well as cylindrical and spherical coordinate systems. Fundamental theorems such as Green's Theorem, Stokes' Theorem and Gauss's Divergence Theorem will also be studied. It provides the foundation for many of the modules available in applied mathematics later in the Honours programme. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT2503 MT2003 2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 5 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework (including class test) = 30% Prof D G Dritschel Prof D G Dritschel

#### Continuous assessment

• Class test (50-minute): 15%
• Fortnightly assessed tutorial questions: 5 x 3% = 15%

#### Syllabus

• Revision of modulus, dot and scalar products (& derivation of cosine formula).
• Grad and directional derivatives of a scalar field; calculation of div and curl of vectors, and curl curl of a vector; verification of identities for div and curl of (scalar times vector); div and curl in cylindrical coordinates; derivatives of unit vectors in spherical coordinates; identity div curl = 0.
• Parametric line integrals in the (x,y) plane; potential function use in line integrals (result depends upon starting and finishing points only).
• Surface integrals of: scalars in spherical coordinates; vectors (in cartesians) using the method of projection; vectors in cylindrical coordinates.
• Green's Theorem in (x,y) with parametric integration; Stokes' Theorem in cartesian and spherical coordinates; Gauss' Divergence theorem in cartesian and cylindrical coordinates.

• Erwin Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, 2011.
• Murray R. Spiegel, Vector Analysis, Schaum’s Outline Series, McGraw-Hill, 1981.
• Robert A. Adams, Calculus: A Complete Course, 6th ed., Pearson Addison Wesley, 2006.
• Robert Adams & Christopher Essex, Calculus, 8th ed, Pearson 2013.

#### MT2507  Mathematical Modelling

Credits 15.0 2 2019/20 12.00 noon Mon (odd weeks), Wed and Fri This module provides an introduction to a variety of techniques that are used throughout applied mathematics. It discusses how to translate physical problems into mathematics and covers such topics as differential equations, dynamics, numerical methods and Fourier series. It illustrates how these are used when solving problems. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT2503 MT2003 2.5 hours of lectures (x 10 weeks), 1-hour tutorial (x 5 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework = 30% Prof T Neukirch Prof T Neukirch

#### Continuous assessment

• Three homework assignment: 3 x 5% = 15%
• Computer project: 15%

#### Syllabus

• Revision of ODEs: separable 1st order ODEs, integrating factors, homogeneous linear 2nd order ODEs with constant coefficients, inhomogeneous linear 2nd order ODEs with constant coefficients. Simple applications: radioactive decay, logistic ODE.  Nonlinear coupled ODEs: application, e.g., predator-prey models, etc., stationary states, linearization.
• Phase plane analysis.
• Dynamics: Newton’s laws, motion under constant gravitational force (1D, 2D), friction, use of total energy.
• Numerical methods: applied to previous nonlinear ODEs, Newton-Raphson (1D, 2D) for calculating stationary states, solution of nonlinear ODEs with numerical methods to supplement phase plane analysis.
• Fourier series: Use of 2D Laplace equation for potential in Cartesian coordinates as motivation, sine and cosine as a system of orthogonal functions, definition of Fourier Coefficients, examples of Fourier series.

• Robert A. Adams, Calculus: A Complete Course, 6th ed., Pearson Addison Wesley, 2006.
• Anton Howard, Irl Bivens & Stephen Davis, Calculus, 9th ed., John Wiley, 2009.
• Erwin Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, 2011.

#### MT2508  Statistical Inference

Credits 15.0 2 2019/20 12.00 noon Mon (even weeks), Tue and Thu This module provides an introduction to the mathematical models of randomness. These models are used to perform statistical analysis, where the aim is to evaluate our uncertainty on a certain quantity after observing data. Important topics in statistics are described including maximum likelihood estimation, confidence intervals and hypothesis testing, permutation tests, and linear regression. It forms a prerequisite for the statistics modules in the Honours programme. It is recommended that students in the Faculties of Arts and Divinity take an even number of the 15-credit 2000-level MT modules. MT2504 MT2004 or EC2003 2.5 hours lectures (x 10 weeks), 1-hour tutorial (x 5 weeks), 1-hour examples class (x 5 weeks) 2-hour Written Examination = 70%, Coursework = 30% Dr H Worthington Dr H Worthington

#### Continuous assessment

• Computer project performing statistical analysis using R: 15%
• Fortnightly assessed tutorial questions: 5 x 3% = 15%

#### Syllabus

• Difference between population and sample: Sample mean and variance as estimates of population mean and variance; sample covariance and correlation.
• Likelihood and maximum likelihood estimation: Discrete data and examples (sequence of binary trials, Poisson counts all with the same mean, Poisson with mean a function of a covariate); continuous data and example (n observations from N(m,s2), m.l.e.s of mean and variance); invariance of m.l.e.s
• Confidence intervals and hypothesis testing: Unbiased and consistent estimators, interval estimation, hypothesis testing
• Basic properties of Normal distributions, Central Limit Theorem (statement and application to binomial and Poisson), assessing normality (normal scores)
• Hypothesis testing and interval estimation for normal distributions with s2 known; c2t and F distributions and their basic properties; one-sample t-test, paired t-test; two-sample t-tests and confidence intervals for means of normal distributions; F-tests for equality of variances of normal distributions; permutation tests: 2-sample permutation test; perm test for matched pairs and one-sample test; randomizaiton tests.
• Simple linear regression: Intro and least squares, normal linear regression, regression in R, CIs and PIs, checking assumptions.

• John A. Rice, Mathematical Statistics and Data Analysis, Belmont, CA: Brooks/Cole CENGAGE, 2007.
• Richard D. De Veaux, Paul F. Velleman & David E. Bock, Stats: Data and Models, Pearson/Addison Wesley, 2005.
• Bryan F.J. Manly, Randomization, Bootstrap and Monte Carlo Methods in Biology, Chapman & Hall, 2007.

#### ID2003  Science Methods

Credits 10.0 1 2019/20 1.00 pm Mon, 1.00 pm Tue, 4.00 pm Thu This module provides an overview of the rationale, methods, history and philosophy of science. We explore the different definitions of science, the distinction between science and pseudo-science, the design of experiments, critical thinking, errors in reasoning, methods of making inferences and generalisations, the role of personal experience and anecdotes in science, the process of scientific publication and the role of anomalies in science. The module is collaboratively taught by staff from a number of schools in the university providing a useful methodological background for all science students. 2x 1-hour lectures (x 11 weeks), 1-hour practical class (x 11 weeks) 1.5-hour Written Examination = 50%, Coursework = 50% Dr C G M Paxton Team Taught

#### ID2005  Scientific Thinking

Credits 15.0 1 2019/20 Lectures: 1.00 pm Mon, Tue, Wed Tutorials: 4.00 pm - 6.00 pm Thu This module provides an overview of the rationale, methods, history & philosophy of science and is a more detailed, 15-credit version of ID2003. We explore the different definitions of science, the distinction between science and non-science, the design of experiments, errors in reasoning, critical thinking, personal experience & science, the grammar of graphics, the process of science, peer review, research reproducibility, data cataloguing, the treatment of anomalies & outliers, and ethics. The module is collaboratively taught by staff from a number of Schools of the University providing a useful methodological background for all science students. ID2003 3 x 1-hour lectures (x 11 weeks), 1-hour tutorials (x 8 weeks), 1-hour seminar (x 1 week) 2-hour practical (x 1 week), 6 hours film/video viewing in total. 1.75-hour Written Examination = 50%, Coursework = 50% Dr C Paxton Team taught