# M.Sc Mathematics ( Sandwich) Courses University of Cape Coast

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MATHEMATICS

Degree Type:

Master of Science

Department:

Department of Mathematics

Programme Duration:

2 years (Standard Entry)

Modes of Study:

Sandwich

About Programme:

In the case of MSc. Programme, without dissertation, students must select courses to attain a minimum of 30 credits of coursework.

The M.Sc programmes are over a twelve-month period involving two semesters of course work.

Objectives

To produce pure mathematics graduates who can undertake research work and create new concepts.

To produce applied graduates who can use mathematics as a tool to do research work in other disciplines such as physics, biology and economics.

To provide a solid foundation for students to pursue advanced and specialised courses in the mathematical sciences.

9 READING LIST

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Diekmann, O. & Heesterbeek, J.A. P. (2000). Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, West Sussex.

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Eisberg, R.M. (2000). Fundamentals of Modern Physics, John Wiley & Sons Inc. New York.

Evans, C. L. (2010). Partial Differential Equations, American Mathematical Society.

Fiacco, A. V. & McCormock, G. P. (1990). Nonlinear Programming, SIAM, Philadelphia, USA.

Fraleigh, J. B. (1989). A First Course in Abstract Algebra.

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Philadelphia, USA.

Gallian, J. A. (1990), Contemporary Abstract Algebra; D. C. Heath and Company.

Gerald, C. F. & Wheatley (2001) Applied Numerical Analysis; Addison &Wesley, USA.

Gibarg, D. & Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order; Springer-Verlag, New York.

Goldstein, H. (1986). Classical Mechanics, Addison-Wesley Publishing Company.

Haaser, N. B. & Sullivan, J. A. (1991). Real Analysis; Dover.

Halmos, P.R. (1960), Measure Theory; Springer-Verlag, New York.

Hertcote , H. W. (2000). The Mathematics of Infectious Disease, SIAM Review, Amsterdam, The Netherlands.

Higham , D. J. (2005). MATLAB Guide, SIAM, Philadelphia, USA.

Hilberland, F. B. (1962). Advanced Calculus for Application; Prentice Hall, USA.

Hillier, F. S. (2012). Introduction to Operations Research, McGraw Hill, Inc., USA.

Hirsch, M. W, Smale, S. & Devaney, R. L. (2004). Differential Equations, Dynamical Systems & An Introduction to CHAOS, Elsevier Academic Press,

Hocking, L. M. (1991), Optimal Control: An Introduction to the Theory with Applications, Clarendon Press, London.

Hungerford, T. W. (1974). Algebra; Springer-Verlag, New York.

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Igor G., Nash, S. G. & Sofer A., (2009). Linear and Nonlinear Optimization, SIAM, Philadelphia, USA.

Kaufmann, J. E. (1987). College Algebra and Trigonometry; PWS Publishers, USA.

Kirk , D. E., (2004), Optimal control theory: An Introduction, Dover Publications.

Klages, R. & Howard, P. (2008), Introduction to Dynamical Systems, (Lecture Notes Version 1.2), Queen Mary University of London.

Kofinti, N. K. (1997). Mathematics Beyond the Basic; Vol. 1, City Printers, Accra.

Kolman, B. (1984). Introductory Linear Algebra with Applications; Macmillan Publishing Company.

Kreyszig, E. (1978 ). Introductory Functional Analysis with Applications; John Wiley and Sons, New York, U.S.A.

Kudryavtsev, V. A. (1981). A Brief Course of Higher Mathematics; Mir Publishers, Moscow.

La Salle, J. P. (1976), The Stability of Dynamical Systems, SIAM, Philadelphia, USA.

Lang, S. (2012). Calculus of Several Variables, Springer-Verlag, New York.

Lenhart S., & Workman J. T., (2007), Optimal Control Applied to Biological Systems, Chapman & Hall, New York, USA.

Lenhart, S., & Workman, J. T. (2007). Optimal Control Applied to Biological, John Wiley & Sons, New York, USA.

Levine, I.N. (1991). Quantum Chemistry, 4th Ed., Prentice Hill.

Levy, A. B. (2009). The Basics of Practical Optimization, SIAM, Philadelphia, USA.

Levy, A. B. (2009). The Basics of Practical Optimization and Control, SIAM, Philadelphia, USA.

Linz, P. & Wang, R. (2002). Exploring Numerical Methods: An Introduction to Scientific Computing Using MATLAB, Jones & Bartlett Publishers, London.

Lipschuts, S. (1975), General Topology; McGraw-Hill Book Company.

Liu, J. H. (2003). A First Course in the Qualitative Theory of Differential Equations, Pearson Education, Inc., New Jersey.

Luenberger, D. G., (1996). Optimization by Vector Space Methods, John Wiley & Sons, New York, USA.

Marion, J.B. & Thornton, S.T. (1995). Classical Dynamics of Particles and Systems, Saunder College Publishers.

Marsden, J.E. (1970). Basic Complex Analysis; W.H. Freeman and Co.

McCann, R. C. Introduction to Ordinary Differential Equations; Harcourt Brace Janovich, USA.

McCoy, N. H. (1968). Introduction to Modern Algebra; Allyn and Bacon Inc.,

Merzbacher, E. (1986). Quantum Mechanics, 2nd Ed. John Wiley & Son Inc.

Morash, R. P. (1987). A Bridge to Abstract Mathematics; Random House Inc., New York.

Munem, M. A. (1989). After Calculus: Analysis; Collier Macmillan Pub. , London.

Nicholson, K. W. (1986). Elementary Linear Algebra with Applications; PWS-KENT.

Ortega, J. M. (1990), Numerical Analysis, SIAM, Philadelphia, USA.

Philadelphia, USA.

Offei, D.N. (1970), The use of boundary condition functions for non-self-adjoint boundary value problems; I

Offei, D. N. (1969). Some asymptotic expansions of a third-order differential equations; Journal of London Mathematical Society, 44 71-87.

Penny, J. & Lindfield, G. (1995), Numerical Methods Using MATLAB, Ellis Horwood, New York.

Petrovsky, I. G.(1954 ). Lectures on Partial Differential Equations; Dover, USA.

Pinchover, Y. & Rubinstein, J. (2005). An Introduction to Partial Differential Equation, Cambridge University Press.

Piskunov, N. (1981). Differential and Integral Calculus; 4 Ed., Mir Publishers, Moscow.

Pliska, S. R. (2002). Introduction to mathematical finance: Discrete time models, Blackwell Publishers Inc.

Poole, D. (2014). Linear Algebra: A Modern Introduction, Dover, USA.

Priestley, H. A. (2003). Introduction to Complex Analysis, 2nd Ed., OUP.

Redheffer, R. (1992). Introduction to Differential Equations; Jones & Bartlett Pub., Inc.

Roberts, A. J. (2009), Elementary Calculus of Financial Mathematics, SIAM, Philadelphia, USA.

Rofman, J. J. (2015). Advanced Modern Algebra, American Mathematical Society.

Roman, S. (2005), Advanced Linear Algebra, 2nd edn; Springer-Verlag, New York.

Ross, S. L. (1984). Differential Equations; 3 Ed., John Wiley & Sons, USA.

Rudin, W. (1974), Principles of Mathematical Analysis; McGraw-Hill Book Company.

Savin, A. & Sternin, B. (2017). Introduction to Complex Theory of Differential Equations, Birkhauser.

Scheid, F. (1988). Numerical Analysis (Schaum Series); McGraw Hill, USA.

Schiff, L.I. (1988). Quantum Mechanics, 3rd Ed., McGraw Hill, New York.

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Smith, K. L. (1988). College Mathematics and Calculus With Applications to Management, Life and Social Sciences; Brooks/Cole Publishing Co., California, USA.

Speyer, J. L. & Jacobson, D. H. (2010). Primer on Optimal Control Theory, SIAM.

Spiegel, M. R. (1991 ). Complex Variables (Schaum Outline Series); McGraw Hill Inc.

Spiegel, M.R. (1992), Real Variables: Lebesque Measure and Integration with Applications to Fourier Series; MacGraw-Hill

Spiegle, M. R. (1991). Advanced Calculus; McGraw Hill, USA.

Stewart J. (1983). Single Variable Calculus; PWS, USA.

Stewart, J. (1987). Calculus; Wadsworth Inc.

Stewart, J. (2003). Calculus (Early Transcendental), 6th Ed. Thomson Brooks/Cole.

Strang, G. (2006). Linear algebra and Its Applications, Thomson Brookes/Cole,

California, USA.

Strauss, W. A. (1992). Partial Differential Equations, an Introduction; John Wiley &

Sons Inc. New York.

Sun, W. & Yuan, Y. (2006), Optimization Theorey and Methods: Nonlinear

Programming, Springer, New York, USA.

Swokowski, E. W. (1984). Calculus With Analytic Geometry; 3 Ed. Prindle, Weber

and Schmidt, Boston, USA.

Symon, K.R. (1973). Mechanics, Addison-Wesley Publishing Company

Systems, Chapman & Hall, New York, USA.

Taylor, A.E. & Lay, D.C. (1988), Introduction to Functional analysis; John Wiley

and Sons.

Teschle, G., (2010). Ordinary Differential Equations and Dynamical Systems:

Graduate Studies in Mathematics, AMS Vol 140, Providence, Rhode Island, USA

Thomas, G. B. and Finney, R. L. (1996). Calculus and Analytic Geometry; 9 Ed.

Addison-Wesley Pub., Reading, USA.

Titchmarsh, E.C. (1972), Eigenfunction expansions associated with second-order

differential equations; Oxford University Press.

Trefethen, L. N., & Bau, D. ( 1997). Numerical Linear algebra ,SIAM, Philadelphia,

USA.

Winston, W. L. (1994). Operations Research: Applications and Algorithms; 3rd Ed.,

Duxbury Press, Belmont, USA.

Winter, R.G. (1986). Quantum Physics, 2nd Ed., Faculty Publishing Inc.

Zill, G. D. (2012). A First Course in Differential Equations with Modelling Applications, John Wiley and Sons.

Entry Requirements:

POST-GRADUATE ADMISSION REQUIREMENT

For the M.Phil/M.Sc in Mathematics, a good first degree in Mathematics, preferably in First Class or Second Class Upper Division, is required.

For the Ph.D in Mathematics, an M.Phil in Mathematics is required.

Career Opportunities:

Programme Structure

Level 100

Second Semester

**MAT 102: ANALYTIC GEOMETRY AND CALCULUS**

3 Credit(s)Pre-requisite: MAT 101

This course is designed to develop the topics of analytic geometry, differential and integral calculus. Emphasis is placed on limits, continuity, derivatives and integrals of algebraic and transcendental functions of one variable. The topics to be covered are: Rectangular Cartesian co-ordinate systems. Distance between two points, gradient of a line, co-ordinates of a point dividing a line segment in a given ratio. Equations of a circle in the form. Points of intersection of lines and circles. Limit of a function of one variable at a point. Continuous functions. Derivatives of a function and its interpretation as the rate of change. Higher order derivatives. Differentiation of algebraic, circular, exponential functions. Sum, product and quotient rules. Differentiation of composite, absolute value and implicit function. Small increments and calculation of approximate values. Application of derivative to increasing and decreasing of functions, maxima and minima. Curve sketching. Integration as the inverse of differentiation. Integration of simple continuous functions and rational functions by substitution. Parametric representation of loci. The parabola, ellipse and rectangular hyperbola. Chords, tangents and normal.

## Level 800

First Semester

**MAT 801: GENERAL TOPOLOGY**

3 Credit(s)Pre-requisite: MAT 430

This course is about the study of properties of topological spaces. Topological spaces turn up naturally in mathematical analysis, abstract algebra and geometry. A topological space is a structure that allows one to generalize concepts such as convergence, connectedness and continuity. Topics covered include: open and closed sets, neighbourhood, basis, convergence, limit point, completeness, compactness, connectedness, continuity of functions, separation axioms, subspaces, product spaces, and quotient spaces.

**MAT 803: FUNCTIONAL ANALYSIS I**

3 Credit(s)Pre-requisite: MAT 408

This course covers major theorems in Functional Analysis that have applications in Harmonic and Fourier, Ordinary and Partial Differential Equations. Topics covered include: Hilbert space as an infinite dimensional generalization of geometric spaces; linear closed subspaces and orthogonality, linear transformations, projections, and spectral theory.

**MAT 805: ORDINARY DIFFERENTIAL EQUATIONS I**

3 Credit(s)Pre-requisite: MAT 405

This course presents the student with advanced techniques for analysing the behaviour of solutions of ordinary differential equations. Topics include systems of first order linear differential equations, existence and uniqueness of solutions; adjoint systems, linear system associated with a linear homogeneous differential equation of order n, adjoint equation to a linear homogeneous differential equation, Lagrange Identity, linear boundary value problems on a finite interval; homogeneous boundary value problems and Green’s function; non-self-adjoint boundary value problems, self-adjoint eigenvalue problems on a finite interval, the expansion and completeness theorems, oscillation and comparison theorem for second-order linear equations and applications.

**MAT 807: MODERN ALGEBRA**

3 Credit(s)Pre-requisite: MAT 403

This course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics. Topics include direct product of groups, finite abelian groups, sylow theorem, finite simple groups, polynomial rings, ordered integral domain, extension fields, algebraic extensions, bilinear and quadratic forms, real and complex inner product spaces, the spectral theory and normal operators.

**MAT 809: ADVANCED TOPICS IN OPERATIONS RESEARCH**

3 Credit(s)Pre-requisite: MAT 409

This course serves as an introduction to the field of operations research. It will quip students with scientific approaches to decision-making and mathematical modelling techniques required to design, improve and operate complex systems in the best possible way. Topics include the formulation of linear programming models: goal programming, transportation problem, case study. Further topics are mathematical programming: project planning and control, dynamic programming, integer programming, probabilistic models: application of queuing theory, forecasting and simulation, decision analysis (making hard decisions), and multi-criteria decision making.

**MAT 811: PARTIAL DIFFERENTIAL EQUATIONS**

3 Credit(s)Pre-requisite: MAT 406

This course focuses on partial differential equation (PDE) models, which will be developed in the context of modelling heat and mass transport and, in particular, wave phenomena, such as sound and water waves. This course develops students’ skills in the formulation, solution, understanding and interpretation of PDE models. Topics include classical theory of partial differential equations together with the modern theory based on functional analysis; systems of partial differential equations, boundary value problems, stability and convergence; applications to the classical problems of mathematical physics.

**MAT 813: MATHEMATICAL EPIDEMIOLOGY**

3 Credit(s)Pre-requisite: MAT 405

This course will examine applications of mathematics in biological contexts including genetics, ecology, physiology, neuroscience and epidemiology. Topics include variants of the MSEIRS epidemic models, disease-free and endemic equilibrium points, determination of the basic reproduction number using the next-generation matrix approach, local stability and global stability analysis of equilibrium points and case studies : HIV/AIDS, TB and Vector-Host Models including Malaria. Further topics are parameter estimation for selected epidemic models, simulation and prediction.

**MAT 815: COMPUTATIONAL LINEAR ALGEBRA**

3 Credit(s)Pre-requisite: MAT 407

This course is an introduction to numerical Linear Algebra. Topics include: matrix factorizations: QR-factorization, Cholesky factorization , vector and matrix norms: properties of the ‖.‖1, ‖.‖2|| and ‖.‖ norms of vectors in Rn, properties of the ‖.‖1, ‖.‖2|| , ‖.‖ and ‖.‖F norms of an mxn matrix, condition number of a matrix, ill-conditioned systems, the Hilbert matrix, perturbation analysis of linear systems, singular value decomposition (SVD) of an mxn matrix, Moore-Penrose inverse, rank k approximation of a matrix, applications of the SVD to least-squares problems, iterative methods for large sparse linear systems: the Jacobi and Gauss-Seidel methods, the SOR method, applications to the solution of linear systems with banded coefficient matrices, regularization methods for ill-conditioned linear systems, regularization of orders 0, 1 and 2, and the L-curve method for choosing an optimal regularization parameter.

Second Semester

**MAT 804: FUNCTIONAL ANALYSIS II**

3 Credit(s)Pre-requisite: MAT 803

This course covers major theorems in Functional Analysis that have applications in Harmonic and Fourier, Ordinary and Partial Differential Equations. Topics covered include: linear spaces, semi-norms, norm, locally convex spaces, linear functional, Hahn-Banach theorem, factor spaces, product spaces conjugate spaces, linear operators, and adjoints.

**MAT 806: ORDINARY DIFFERENTIAL EQUATIONS II**

3 Credit(s)Pre-requisite: MAT 805

This course presents the student with advanced techniques for analysing the behaviour of solutions of ordinary differential equations. Topics include linear systems with isolated singularities, linearisation of systems of differential equations, asymptotic behaviour of non-linear systems: stability, perturbation of systems having a periodic solution, perturbation theory of two-dimensional real autonomous systems.

MAT 808: BOUNDARY CONDITION FUNCTIONS

3 Credit(s)Pre-requisite: MAT 405 and MAT 406

This course introduces students to the construction of Green’s functions for boundary value problems. Topics include boundary condition functions for self-adjoint and non-self-adjoint boundary value problems, construction of Green’s functions in terms of boundary condition functions, aymptotic behaviour of boundary condition functions and Green’s functions, and singular self-adjoint boundary value problem.

**MAT 810: COMPLEX ANALYSIS**

3 Credit(s)Pre-requisite: MAT 404

This course provides advanced topics in complex analysis such as conformal mappings, and physical applications of conformal mapping. Further topics include analytic continuation and Riemann surfaces, Rouches’s theorem and principle of the argument, mapping properties of analytic functions, inverse function theory, maximum modulus theory, infinite products and gamma function, Sterling’s formula and Bessel’s functions, homotopy and homology theory, and analytic properties of holomorphic functions.

**MAT 812: IMAGE PROCESSING**

3 Credit(s)Pre-requisite: MAT 815

This course introduces the basic theories and methodologies of digital image processing. The topics include manipulating images in MATLAB/OCTAVE, images as Arrays of Numbers, digital image, compression the singular value decomposion, the image de-blurring problem: a mathematical model of the blurring process. Further topics include de-blurring using a general linear model, obtaining the point spread function (PSF). De-bluring images using TSVD method, total variation method, and the Tikhonov regularization method, general image reconstruction as an inverse problem.

**MAT 814: OPTIMIZATION**

3 Credit(s)Pre-requisite: MAT 301 and MAT 305

This course introduces the principal algorithms for linear, network, discrete, nonlinear, dynamic optimization and optimal control. Emphasis is on methodology and the underlying mathematical structures. Topics include unconstrained optimization: optimality conditions, Newton’s method, quasi-Newton’s methods, Steepest Descent Method, Conjugate-Gradient methods, Line Search methods, Trust Region Methods, Derivative-Free Methods, constrained Optimization: optimality conditions for (a) linear equality constraints, (b) linear inequality constraints, (c) nonlinear constraints, feasible-point methods, sequential quadratic programming (SQP), reduced-gradient method, penalty and barrier methods.

**MAT 816: OPTIMAL CONTROL**

3 Credit(s)Pre-requisite: MAT 814

This course investigates how dynamical systems should be controlled in the best possible way. Topics include: OCP with bounded and unbounded controls. Bang-Bang controls, Singular controls. OCPs with linear and nonlinear dynamical systems. OCPs for systems with fixed or free terminal times. OCPs for systems with equality and inequality constraints on functions of state and control variables. Numerical Methods for OCPS: Control parametrization method, State Discretization methods, Lenhart’s Forward-Backward Sweep method. Application to the conrol of dynamical systems, including the control of infectious diseases.

**MAT802: MEASURE AND INTEGRATION**

3 Credit(s)Pre-requisite: MAT 408 and MAT 801

This course covers advanced topics in abstract measure theory and Lebesgue integration. Topics covered include: measurable sets and functions, measure spaces, Lebesgue integral, monotone convergence theorem, Fatou’s lemma, Lebesgue dominated convergence theorem, Vitali’s theorem, decomposition of measures, Caratheordory and Hahn extension theorem, spaces, Riesz representation theorem, and product measures.