Applied Mathematics (BSc)
These regulations shall be read in conjunction with the General Regulations and the Faculty Regulations.
2. AIMS AND OBJECTIVES
After completion of the degree programme students should be able to:
– apply mathematical and economic concepts, principles and processes in the areas of Science, Engineering, Industry and Commerce
– pursue post- graduate studies in mathematics and related areas
– design and implement a computer programming task as well as a sound knowledge, of mathematics and of business studies across a broad spectrum of areas.
– teach secondary school mathematics with computer applications up to `A’ Level and
– conduct research at whatever level.
3. EMPLOYMENT PROSPECTS
Applied Mathematics has immense applications in the fields of Science, Engineering, Industry and Commerce. After students graduate they are employable in the following areas: Acturial Sciences; Financial Institutions: (banks, building societies, insurance companies and pension funds); Meteorology (forecasting); Manufacturing industry (production/operation management); Industrial Research; Research and Project Management in NGOs, research stations, etc; Mining Sector; Teaching and Lecturing and many other relevant areas of the economy.
4. ENTRY REQUIREMENTS
4.1 Normal Entry
To qualify for the programme, applicants must meet the general entry requirements, and in addition they should have passed “A” Level Mathematics and any other science or commercial subject or their recognized equivalents.
4.2 Special Entry
To qualify for special entry, applicants must have a diploma in Mathematics or any other related field.
4.3 Mature Entry
Refer to Section 3.3 of the General Regulations.
5. GENERAL PROVISIONS
Refer to Section 4 of the Faculty Regulations
Assignments and tests will be given for each module that a student takes. The continuous assessment mark C = 0.25 (assignment average) + 0.75 (test average).
For each module a student will write a two hour examination at the end of the semester and the final mark for each module is F = 0.25C + 0.75E, where E is the examination mark.
7. FAILURE TO SATISFY EXAMINERS
Refer to section 9 of the General Regulations.
8. PROVISION FOR PROGRESSION
Refer to Section 7 of the Faculty regulations.
9. WORK RELATED LEARNING GENERAL GUIDELINES
Refer to Section 10.2 of the General Regulations.
10. GRADING AND DEGREE CLASSIFICATION
Refer to Section 5 of the General Regulations.
11. DEGREE WEIGHTING
Refer to Section 11 of the Faculty Regulations.
12. PROGRAMME STRUCTURE
N.B Modules marked * are core modules.
Level 1 Semester 1
|HMT101||* Calculus I||4|
|HMT102||* Linear Mathematics I||4|
|HMT103||*Probability Theory I||4|
|HMT104||* Applied Statistics||4|
|HCS101||Introduction to Computers and Computer applications||4|
|HCS103||Digital Logic Design||4|
Level 1 Semester 2
|HMT105||* Mathematical Discourse And Structures||4|
|HMT106||Calculus II ( HMT101)||4|
|HMT107||Probability Theory II (HMT103)||4|
|HMT108||* Regression And Anova I (HMT104/ HMT103)||4|
|HCS102||Introduction To Programming||4|
|HCS104||Systems Analysis and Design||4|
Level 2 Semester 1
|HMT201||* Ordinary Differential Equations (HMT101)||4|
|HMT202||Linear Mathematics II (HMT102)||4|
|HMT203||* Statistical Inference I (HMT103/HMT104)||4|
|HMT 205||Vector Calculus||4|
|HMT211||Time Series Analysis||4|
|HMT209||Mathematical Finance 1||4|
|HCS204||Data Comms & Computer Networks||4|
|GS201||* Introduction to Gender Studies||4|
Level 2 Semester 2
|HMT212||* Real Analysis I HMT101||4|
|HMT210||Operations Research I||4|
|HMT204||* Design And Analysis of Experiments||4|
|CT211||Quality Managements Systems||4|
|HCS206||Models Of Database & Database Design||4|
Level 3 Semester 1and 2
|HMT301||Work-Related Learning Report||15|
|HMT302||Academic Supervisor’s Report||15|
|HMT303||Employer’s Assessment Report||10|
Level 4 Semester I
|HMT401||* Pdes and Fourier Analysis||4|
|HMT402||* Linear Models HMT108||4|
|HMT403||Mathematical Finance II HMT209||4|
|HMT405||Hypothesis Testing HMT103||4|
|HMT406||Real Analysis II HMT212||4|
|HMT408||Non-Linear O.D.E S||4|
|HMT409||Statistical Inference II HMT203||4|
|HMT421||Theory of Estimation||4|
Level 4 Semester 2
|HMT414||* Mathematical Modelling||4|
|HMT415||Economic and Social Statistics||4|
|HMT418||Regression and ANOVA II HMT108||4|
13. MODULE SYNOPSES
HMT101 CALCULUS I
Number systems: Natural, integral, rational and irrational. The principle of mathematical induction. The real number system: decimal and geometrical representation, inequalities and their solution sets. Functions: exponential, logarithmic, circular and hyperbolic and their inverses. Limits of functions. Continuity. Sequences: convergence of a series as convergence of the sequence of partial sums. Differentiation: Derivatives of functions of a single variable. Integration: The definite integral, the indefinite integral or antiderivative, practical techniques of integration, method of substitution, integration by parts and reduction formulae, fundamental theorem of calculus.
HMT102 LINEAR MATHEMATICS I
Complex numbers: geometric representation, algebra. De Moivres theorem polynomials and roots of polynomial equations. Matrices and determinants: algebra of matrices, inverses, definition and manipulation of determinants, solutions of simultaneous linear equations, applications to geometry and vectors. Differential equations: separable, homogeneous, exact, integrating factors, linear equation with constant coefficients.
HMT103 PROBABILITY THEORY I
Axiomatic probability, sets and events, sample space, conditional probability, Independence, laws discrete and continuous random variables, probability density functions, mean, variance, expectation. Independence, Chebyshev’s inequality, moments and moment generating functions. Common Discrete Distributions, Uniform, Bernoulli and Binomial, multinomial, hypergeometric, Poisson, Geometric and negative binomial. Use of tables. Common Continuous Distributions: Uniform, Normal, Exponential, gamma, beta. Use of tables. Joint Probability Distributions. Conditional and marginal distribution, expectation, covariance and correlation. Approximations, Law of large numbers, Central limit Theorem, Normal approximation to binomial, poisson, etc.
HMT104 APPLIED STATISTICS
Graphical techniques. Kinds of. Measures of central tendency. Measures of variability. Empirical distributions. Moments. Skewness and Kurtosis. Applications. Indicators. Contingency Tables. Introduction to Time Series trends. Sampling .Introduction to estimation procedures: Judgemental method and Method of moments. Introduction to Hypothesis testing. Ideas about non-parametric statistics. Chi-square contingency methods, Goodness of fit, Q-Q plots, using applications in agricultural and health statistics.
HMT105 MATHEMATICAL DISCOURSE AND STRUCTURES
Sets: formulae, propositions, Boolean Algebra and its applications. Logic, mathematical reasoning and proof: examples taken from various areas of mathematics. Relations: binary, n- ary, reflexive, symmetric, transitive, equivalence relations and classes, partitions, order relations, inverse relations. Functions: one to one, onto, inverse functions. Operations: sets with one or two binary operations: permutations, symmetry groups, modular arithmetic, etc.
HMT106 CALCULUS II
Theorems on differentiation, higher order derivatives and Leibnitz,s formula. The Mean value theorems: Rolles theorem, the Mean value theorem, the generalised mean value theorem, Taylor s theorem. Applications to maxima and minima, curve sketching, approximations and Newton’s Method. Leibnitz’s Theorem, Functions of several variables: limits, continuity. Differentiation of functions of several variables, Tailors theorem. Applications of maxima and minima problems, Lagrange multipliers. Multiple and triple integrals: change of order of integration transformations, Applications to finding area, volume, arc length, centroid, moments of inertia, etc. Series: tests of convergence, absolute and conditional convergence , series of functions, uniform convergence.
HMT107 PROBABILITY THEORY II
Bivariate probability distributions. Moment generating functions. Characteristic functions. Multinomial and multivariate normal distributions. Distributions of functions of random variables. Cumulative distribution function technique. Expectations of functions of random variables. The transformation Y=g(x). Probability integral transform, simulation of random numbers. Other transformations for discrete and continuous random vectors. Sampling distributions. Law of large numbers
HMT108 REGRESSION AND ANALYSIS OF VARIANCE I
Correlation and regression, scatterplots, correlation matrix. Method of least squares, associated lines, assumptions underlying regression. Checking validity of assumptions. Residuals and transformations. Outliers. Pearson’s and Spearman’s correlation coefficients, predictions. Regression in terms of sums of squares and sums of products. Estimation and testing, t and F-tests. Multiple linear regression: linear equations and matrices. Matrices in simple and multiple linear regression. Testing and inference in multiple linear regression using matrices. Partial correlation. Analysis of variance (ANOVA). Assumptions underlying ANOVA. One-way, balanced design ANOVA.
HMT201 ORDINARY DIFFERENTIAL EQUATIONS
Basic techniques for solution of first and second order differential equations. Method of undetermined coefficients and method of variation of parameters. Existence and Uniqueness of solutions. Series solution. Differential equations of special functions. Laplace transforms to the solution of ODEs.
HMT202 LINEAR MATHEMATICS II
Vector spaces, linear dependence and independence, bases and dimension. Linear transformations, operations on linear operators. Eigenvectors, eigenvalues, orthogonality of eigenvectors, geometric and algebraic multiplicity of eigenvalues. Applications of diagonalisation of matrices, quadratic and bilinear forms, Jordan Normal form of a matrix, solution of systems of differential equations. The Cayley Hamilton Theorem and its applications.
HMT203 STATISTICAL INFERENCE I
Deductive inference, population and sample concepts as the basis of statistical inference, parameters and statistics, review of probability theory. Central Limit Theorem, Chi-square, student-t and F distributions, distribution of min and max. Estimation: methods of estimation, properties of estimators and their sampling distributions. Interval estimation. Confidence intervals. Hypothesis testing.
HMT204 DESIGN AND ANALYSIS OF EXPERIMENTS
Principles of experimentation. Randomisation, replication, treatment structure, blocking and error control. The two-way model; fixed and random effects. Randomised block designs; Latin Square designs; Balanced Incomplete Block Designs; Crossover Designs. Checking model assumptions. Techniques for missing observations. Balance and Orthogonality. Analysis of Covariance, Factorial designs; fractional factorials; confounding. Split plot designs. Repeated Measures. Nested designs. Ideas of response surface methodology.
HMT205 VECTOR CALCULUS
Brief review of line, surface and volume integrals and applications to work done, flux through surfaces. Grad, curl and Stokes’ Theorem: characterization of conservative vector fields, existence of potential functions in simply connected domains. Div and divergence theorem: solenoidal vector fields and vector potential. Comparison of integral theorems with the Fundamental theorem of calculus; derivation of the continuity equation. The Laplacian in polar, cylindrical and spherical coordinates. Field lines, sources and sinks. Gauss’s law for continuous distributions of sources; Poisson’s equation and its solution for some simple geometries.
HMT206 NUMERICAL METHODS
Introduction to simple numerical methods for solving problems in Mathematics, Science and Finance. Computer arithmetic and rounding errors. Numerical Methods for root-finding, simple iterative methods and the Newton-Raphson method, convergence. Polynomial interpolation and splines. Solution of linear algebraic equations, scaled partial pivoting. Numerical integration and differentiation. Numerical integration of ODEs. Euler and second order. Runge-Kutta methods.
HMT207 SURVEY TECHNIQUES
Uses, scope and advantage of sample surveys. Types of surveys. The phases of a survey. Survey organisation. Questionnaire design, dummy tables, pre-tests, training of field workers. Report writing. Errors in surveys, monitoring reviews, quality control.
Sample design. Further sampling theory. Estimation of means, totals, proportions. Ratio estimation. Variance calculations. Practical work.
HMT 208 NUMBER THEORY
Foundations: Integers, well-ordering principle, induction, Fibonacci numbers; Divisibility; prime numbers, distribution of primes, conjectures about primes; greatest common divisor, least common multiple; Euclidean algorithm; fundamental theorem of arithmetic and applications, Dirichlet progressions, irrational numbers; Fermat factorization; linear Diophantine equations; perfect numbers, Mersenne numbers; Congruences: linear congruences; Chinese remainder theorem; Wilson’s and Fermat’s little theorem; primality testing and Carmichael numbers; Euler’s theorem; properties of the Euler Phi function; sum and number divisors; Moebius inversion; Public-key cryptography, RSA encryption method, knapsack ciphers; Digital signatures, Diffie-Hellman key exchange
HMT 209 MATHEMATICAL FINANCE I
Theory of interest rates: simple interest, compound interest, nominal rates of interest, accumulation factors, force of interest, present values, Stoodley’s formula for the force of interest, present values of cash flows. Basic compound interest functions: the equation of value and yield of a transaction, annuities certain, present values and accumulations, deferred annuities, continuously payable annuities, the general loan schedule, the loan schedule for a level annuity. Nominal rates of interest: annuities payable pthly, annuities payable pthly, present values and accumulations, annuities payable at intervals of time r, where r>1, the loan schedule for a pthly annuity. Discounted cash flow: Net cash flows, Net present values and yields, comparison of two investment projects, different interest rates for lending and borrowing, effects of inflation, the yield of a fund, measurement of investment performance. Capital redemption policies: Introduction to premium calculations, policy values, policy values when premiums are payable pthly, surrender values, paid up policy values and policy alterations, variations in interest rates, Stoodley,s logistic model for the force of interest, reinvestment rates.
HMT 210 OPERATIONS RESEARCH I
Review of mathematical programming methods: linear programming, simplex method, M-technique, Dual linear programming methods. Dynamic programming: problem formulation and solution. Project scheduling: network construction, PERT-CPM methods, project control. Queuing theory: single-queuing models (MM/1), Multi-server queuing models (MM/c), finite queue variation and P. K. formula. Inventory Control Models: Optimator economic order quantities, deterministic models for single and multiple items. Probability models. Decision Analysis: Bayesian methods, minimax-maxi-maxi criteria, maximum likelihood, maximal opportunity criteria, introduction to Utility Theory.
HMT 211 TIME SERIES ANALYSIS
Time series models, estimation and elimination of trend and seasonal variation. Tests of randomness and normality. Introduction to projects. Model building strategy. Variance and covariance of linear combinations. Time series as a stochastic process, stationary stochastic process, white noise, and random walk. Variance of sample mean estimation of trends and seasonal variation. Sample ACP. Variance of sample autocorrelation and corresponding significance test for zero autocorrelation. General linear process. Autocovariance generating function. Moving average process. Invertibility. Autoregressive processes. Yule-Walker equation. Solution of difference equations. AR(1) and AR(2) processes: stationarity conditions. ARMA (1,1) process. General ARMA(p,q) process. ARIMA models for non-stationary processes: IMA(1,1), AR(1,1) IMA(2,2) models. Log transformation to stationarity. Identification of ARIMA models. Var (Z) for stationary processes. Partial ACF and applications to AR(1), AR(2) and MA(1) models. Parameter estimation: method of moments, least squares, and maximum likelihood. Properties of parameter estimators. Goodness of fit: Box-Pierce statistics, overfitting, autocorrelation of residuals. Forecasting: minimum mean measure square error forecast; forecast errors. Applications to ARMA and ARIMA processes. MA(1)12, AR(1) 12, ARMA(1,1) 12 models. Multiplicative seasonal ARMA(p,q) x (P,Q)s model. Introduction to the frequency domain. Periodogram and spectral analysis.
HMT 212 REAL ANALYSIS I
Historical development of the real number system. Countability, cardinal numbers, existence of transcendental numbers. The real numbers as a complete, ordered field. Supremum axiom, Archmedean property, principle on monotone bounded convergence, nested interval theorem, Bolzono-Weierstrass theorem, convergence of Cauchy sequences. Limits and continuity of real functions. Boundedness theorem, intermediate value theorem, interval theorem, application to fixed point theorem. Uniform continuity. Differentiability. Local extrema, Rolle’s theorem, mean-value theorem, L’Hospital’s rule, Leibuiz’s theorem, Taylor’s theorem. Applications to finding roots, curve sketching, classification of local extrema and approximation by polynomials. The Riemann integral, integrability, properties of the Riemann integral, the mean-value theorem for integrals, the fundamental theorem of calculus.
HMT 401 PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES
Partial differential equations of mathematical physics and economics. Classification of second order PDEs in two independent variables. Derivation of the wave, Laplace and Poisson equations, method of separation of variables and Laplace techniques. Orthogonal sets of functions in an inner product space. Introduction to Hilbert spaces. Fourier Series. Fourier sine and Fourier cosine series. Discussion of a convergence theorem. Integration and differentiation of Fourier series to boundary value problems. Fourier series in two variables. The Fourier transforms and it’s inverse. The convolution theorem.
Applications: Bessel functions J(x). The zeroes of J(x). orthogonal sets of Bessel functions. Fourier-Bessel series. Applications of the theory to the solution of PDEs will be stressed throughout.
HMT 402 LINEAR MODELS
Regression: Linear regression model, point and interval estimation of parameters. Pure error and lack of fit. Residual analysis. Multiple regression: estimation and confidence intervals. General linear hypothesis. Stepwise methods. Experimental design models: one factor models. Fixed and random effects. Two factor models, with and without interaction. Qualitative and quantitative contrasts.
HMT 403 MATHEMATICAL FINANCE II
Introduction to the mathematical models used in finance and economics with emphasis on pricing derivative instruments. Financial markets and instruments; elements from basic probability theory; interest rates and present value analysis; normal distribution of stock returns; option pricing; arbitrage pricing theory; the multiperiod binomial model; the Black-Scholes option pricing formula; proof of the Black-Scholes option pricing formula and applications; trading and hedging of options; Delta hedging; utility functions and portfolio theory; elementary stochastic calculus; Ito’s Lemma; the Black-Scholes equation and its conversion to the heat equation.
HMT 404 OPTIMIZATION
Basic results, conditions for unconstrained variables, equality constraints, inequality constraints, duality, unconstrained optimization, linear programming, constrained optimization.
HMT 405 HYPOTHESIS TESTING
Introduction to testing of hypothesis. Simple hypothesis versus simple alternative. Composite hypothesis. Sampling from the normal distribution. Chi-square tests. Tests of equality of two multinomial distributions and generalizations, tests of independence in contingency tables. Sequential tests of hypothesis.
HMT 406 REAL ANALYSIS II
Riemann Integral: Definition, Darboux’s theorem, Riemann sums, fundamental theorem. Uniform convergence of sequences and series of functions, power series. Lebesgue Measure: Borel ó-algebra, outer measure, Lebesgue measurable sets, Lebesgue measure and its properties. Lebesgue Integration: simple functions, measurable functions. Monotone convergence theorem, dominated convergence theorem, and relation between Lebesgue and Riemann integral.
HMT 408 NON-LINEAR ODER
Second order differential equations in the phase plane. Plane autonomous systems and linearization. Geometrical aspects of plane autonomous systems. Periodic solutions; averaging methods. Perturbation methods. Singular perturbation methods. Forced oscillation; harmonic and subharmornic response, stability and entrainment. Stability. Determination of stability by solution perturbation. Liapunov methods for determining stability of the zero solution. The existence of periodic solutions. Bifurcations and manifolds. Poincare sequences, homoclinic bifurcation and chaos.
HMT409 STATISTICAL INFERENCE II
Types of statistical data. Order statistics. Exact and asymptotic distribution of order statistics. Review of sign median, run tests. Mann-Whitney U statistics; definition, use. Asymptotic mean and variance. Wilcoxon one-sample and two sample tests. Testes of location vs. tests of variability: Siegel-Tukey or Ansari-Bradley, Waish. Non-parametric tests for experimental design: Kruskai-Wallis, Friedman, Durbin. Test for ordered alternatives: Jonckheere-Terstra, Page. Test for extreme reactions. Hollander. Tests for dichotomised or cardinal data: Cochran, McNemar. Kendal’s measures. Fisher’s exact test. Chi-square based tests. Other goodness-of-fit tests; Koimogorov’-Smirnov, generation of uniformly distributed random numbers, generation of random numbers by inverse transform, other methods of generations. Bootstrap and Jackknife estimation. Resampling. M-, L- and R-estimator.
Introduction to econometrics, its role in Zimbabwe. Review of general linear model: least squares estimators, correlation matrix, partial correlation coefficients regression coefficients, Significance tests and confidence intervals. Linear restrictions, multicollinearity, specification error, dummy variables. Generalised least squares, GLS estimator, heteroscedasticity, pure and mixed estimation, group observations and grouping of equations. Autocorrelation: sources consequences and conventional tests including Durbin-Watson, Thiel BLUS procedure, estimation, and prediction. Stochastic regressors: definition, instrumental variables, errors in variables. Lagged variables: lagged explanatory variables, lagged dependent variable and estimation including Koyck’s lag Scheme. Simultaneous equations systems, endogenous and exogenous variables identification problem. Restrictions on structural parameters. Two stage and three stage least squares.
HMT411 GRAPH THEORY
Introduction to the abstract known as graph. Definitions and characterization of classes of special graphs. Distance and connectedness measures. Various algorithms applied to graphs and some of their proofs, classical and contemporary.
HMT412 FLUID MECHANICS
Introduction to the nature of fluids. Hydrostatics and pressure. Equation of motion and Bernoulli’s equation. Vorticity and circulation, Kelvin’s theorem. Two dimensional flow, velocity potential and stream function, complex variables. Axisymetric flows and Stokes’ stream function. Unsteady flows with vorticity. Applications of fluid mechanics in meteorology, engineering, etc.
HMT413 MATHEMATICAL PROGRAMMING
Introduction to mathematical programming problems: Linear programming problem formulation; simplex method; Chmens and two phase techniques; sensitivity analysis; duality in LP; Dual simplex method; transportation and assignment methods; integer programming; dynamic programming; quadratics and separable programming; K T conditions for optimality.
HMT414 MATHEMATICAL MODELING
Aims and Philosophy of Mathematical Modeling. Modeling methodology, role and limitations. Mathematical modeling with sequences and series. Differential calculus, first order differential equations. Population and ecological models: application of linear autonomous systems to the physical and biological sciences. Optimal Policy Decisions: Models based on optimization techniques. Case studies.
HMT415 ECONOMIC AND SOCIAL STATISTICS
Economic principles, national accounts, financial and industrial statistics, price indices, international trade, balance of payments. Agricultural statistics, Demographic concepts and measures, methods of population enumeration, cohort and life tables. Uses of demographic data, socio-economic indicators, labour, health, education, social welfare, women and men, crime, household surveys etc
HMT416 STOCHASTIC PROCESSES
To include integer-valued variables: probability generating functions, convolutions. Markov chains: transition probabilities, classifications of states, stationary distributions, transient states. Gambler ruin, random walk. Markov processes: Chapman-Kolmogorov equations, transition rate matrix, forward and backward systems. Poisson process, normal equations, machine operation machinery breakdown, queuing model.
HMT417 MULTI VARIATE ANALYSIS
Multivariate data, descriptive statistics, graphical techniques. Random vectors and matrices, their expectations and properties, expected values of the sample mean and sample covariance matrix. Multivariate normal distribution and its properties, sampling distribution of sample mean and sample covariance matrix. Wishart distribution. Transformation to near normality, testing for normality. Inferences about mean vector: Hotelling’s T^2 distributions and likelihood ratio test, comparison with one dimensional case, confidence regions and simultaneous comparisons of component means: Bonferroni method of multiple comparison. Comparison of several multivariate means: comparing mean vectors from two population means (one-way MANOVA). Simultaneous confidence intervals for treatment effects, profile analysis, ideas of two-way MANOVA. Review of eigenvalues and vectors, spectral decomposition of symmetric matrix. Principal component analysis. Introductory study and use of one technique from; Factor analysis, Canonical correlation analysis, Discriminant analysis.
HMT418 REGRESSION AND ANALYSIS OF VARIANCE
Multivariate normal distribution. Matrix and vector algebra. Linear model in matrix notation. Fitting a straight line by least squares, lack of fit, pure error, review of multiple linear regression. Estimation, testing hypotheses and confidence regions for parameters in full rank linear regression model, polynomial regression models. Regression diagnostics. An introduction to stepwise regression applied to analysis of variance problems. The one-way and two-way classifications. An introduction to non-linear regression.
Kinematics, projectiles, Newton’s Laws, forces, momentum, work, energy, power, conservative and dissipative forces. Orbits. Oscillations, elastic forces and resonance. Equivalent systems of forces plane statistics, system of particles, elementary theory of rigid bodies.
HMT 420 COMPLEX ANALYSIS
Complex numbers, elementary operations: addition, multiplication, their properties. The conjugate, the absolute value and their behaviour with respect to addition and multiplication. Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses. Open, closed, connected sets. Limits of sequences and functions, their behaviour with respect to addition, multiplication, division. Cauchy’s criterion for convergence. Continuous functions. Continuity of sums, products, ratios, compositions. Definitions of continiuty using open and closed sets. Connectedness, its preservation under continuous maps. Uniform convergence and continuity of uniform limits of continuous functions. Branches of multi-valued functions. Examples of branches of the argument function and the logarithm. Infinite series of complex numbers. Geometric series and its convergence properties. The comparison test. Absolute convergence. Infinite function series and their uniform convergence. Weierstrass test. Power series. Abel’s Lemma. Radius of convergence. Complex-differentiable and holomorphic functions. Differentiability of sums, products, ratios, composition and inverse functions. Real-differentiable functions. Cauchy-Riemann equations. Complex differentiability of polynomials, rational functions, exponential, logarithm and trigonometric functions. Path integrals. Independence of parametrization. Length of a path and estimates for path integrals. Antiderivatives. Calculation of path integrals using antiderivatives. Cauchy’s theorem: Goursat’s version for a triangle, generalization for polygonal regions and simple bounded regions. Cauchy’s integral formula. Residue theorem. Calculation of residues for ratios of holomorphic functions. Applications of Residue theorem: Trigonometric integrals, Improper integrals, Fourier transform type integrals etc. Taylor series and Laurent series expansions. Differentiation of power series. Poles. Calculation of residues using Laurent series expansion. Order of zeroes and poles. Identity principle. Maximum modulus principle.
HMT 421 THEORY OF ESTIMATION
General Minimum Variance Unbiased Estimation, Cramer-Rao Lower Bound, Linear Models & Unbiased Estimators, Maximum Likelihood Estimation, Least squares estimation, Bayesian Estimation, Statistical Detection Theory, Deterministic Signals, Random Signals, Non-parametric and robust detection
HMT422 ABSTRACT ALGEBRA
Groups with operators, homomorphism and isomorphism theorems, normal series, Sylow theorems, free groups, Abelian groups, rings, integral domains, fields, modules. Topics may include HOM (A,B), Tensor products, exterior algebra.
The student undertakes a project based on a thorough study of some mathematical and/or statistical aspects of the theory with reference to applications. The project will be supervised by a departmental lecturing member of staff and should therefore be of high standard.
GS201 Gender Studies
Refer to the Department of Gender Studies
CS101 Communication Skills
Refer to the Department of Communication Skills
HCS101 Introduction to Computers and Computer Applications
Refer to the Department of Computer Science
HCS103 Digital Logic Design
Refer to the Department of Computer Science
HCS102 Introduction to Programming
Refer to the Department of Computer Science
HCS104 Systems Analysis and Design
Refer to the Department of Computer Science
HCS203 Operating Systems
Refer to the Department of Computer Science
HCS204 Data Communications and Computer Networks
Refer to the Department of Computer Science
HCS206 Models of Database and Database Design
Refer to the Department of Computer Science
Refer to the Department of Computer Science
CT211 Quality Management Systems
Refer to the Department of Chemical Technology