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Elementary set theory, subsets, union, intersections, complement, Venn diagrams. Real numbers; integers, rational and irrational numbers, mathematical induction, real sequences and series, theory of quadratic equations, binominal theorem, nth roots of unity. Circular
measure, trigonometric functions of angles of any magnitude, addition and factor formulae.
Function of a real variable, graphs, limits and idea of continuity. The derivative, as limit of rate of change. Techniques of differentiation. Extreme curve sketching, Integration as an inverse of differentiation. Methods of integration, Definite integral. Applications areas, volumes areas, etc.
Geometric representation of vectors in 1-3 dimensions, components, direction cosines. Addition, Scalar, multiplication of vectors, linear independences, Scalar and scalar variable. Two dimensional co-ordinate geometry. Straight lines. Circles, parabola, ellipse, hyperbola Tangents, normal, Kinematics of a particle, components of velocity and acceleration of a particle moving in a plane force, momentum, laws of motion, under gravity projections,
resisted vertical motion, elastic string, simple pendulum, impulse. Impact of two smooth sphere and of a sphere on a smooth sphere. Pre-requisite – MTS 101.
Measure of location and dispersion in simple and grouped data, Elements of probability and probability distribution, normal, binomial, Poison, geometric, Negative binomial distribution. Estimation and tests of hypothesis concerning the parameters f-distribution. Regression,
correlation and analysis of variance, contingency table Non-parametric inference.
Real valued function of a real variable. Review of differentiation and integration and their applications, Mean value theorem, Taylor series, Real-value functions of two or three variable. Partial derivatives, chain rule, extrema, nLagrange’s multipliers. Increments, differentials and linear approximations. Evaluation of linear integral. Prerequisites MTS 102
Solution to algebraic and transcendental equations. Curve fitting. Error analysis, Interpolation and approximation. Zeros of non-linear equations of one variable, System of linear equations. Numerical differentiation and integration.
Vector space over the real field, Subspaces, linear independence, basis and dimension. Linear transformation and their representation by matrices-range, null space, rank. Singular and non-singular transformation and matrices. Algebra of matrices. Pre-requisite: MTS 101
Systems of linear equation, change of basis, equivalenceand similarity. Eigenvalues and eigenvectors. Minimum and characteristics polynomials of a linear transformation (Matrix) Cayley Hamilton’s theorem. Bilinear and quadratic forms, orthogonal diagonolisation. Canonical forms. Pre-requisite MTS 104 and IMT 101.
Bounds of real numbers, convergence of a sequence of numbers. Monotone sequences, the theorem of nested intervals, cauchy sequences, and test for convergence of a series. Absolute and conditional convergence of a series, and rearrangements. Completeness of reals and
incompleteness of rational. Continuity and differentiability of function (R…) Rolle’s and mean value theorems for differentiable functions. Taylor series.
System of linear Equations, change of basis, equivalence and similarity, Eigen-values and eigen-vectors, minimum and characteristics polynomials of a linear transformation (matrix), Caley Hamilton’s theorem. Bilinear and quadratic forms orthogonal diagonalization canonical
forms
Firs order ordinary differential equations. Existence and uniqueness. Second order ordinary differential equations with constant coefficients. General theory of nth order linear equations. Laplace transforms, solution of initial value problems by laplace transform method.
Simple treatment of partial differential equations in two independent variables. Application areas. Prerequisite MTS 102.
Vectors in Euclidean Spaces. Dot ad Vector products. Element of Vector Calculus. Gradients of Scalar functions. Curl and Divergence of Vector fields. General Kinematics, momentum, angular momentum, fundamental equations of motion. Energy and conservation laws. Particle and rigid body Dynamics. Simple Harmonic oscillators and simple pendulum.
Combinatorial analysis. Probability models for the study of random phenomena in finite sample spaces. Probability ndistributions of discrete and continuous random variables. Expectations and moments generating function, Cebyshev’s inequality. Bivariate, marginal and
conditional distributions and moments. Convolution of two distributions, the central limit theorem, and its uses. Pre-requisite MTS 101.
Use of statistical methods in biology and agriculture. Frequency distributions. Laws of probability. The binominal. Poison and normal probability distributions. Estimation and tests of hypothesis. Design of simple agricultural and biological experiments. Analysis of variance and covariance, simple regression and correlation,contingency tables. Some non-parametric tests.
Time series data Analysis, Fitting parametric models, Regression and auto regression models, ARCMA, Stationary and Non- stationary models. Special analysis filtering.
Vector algebra. Vector, dot and cross products. Equation of curves and surfaces. Vector differentiation and applications. Gradients. Divergence and curl. Vector integration, Line surface and volume integrals. Green’s, stokes’ and divergence theorems. Tensor products of vector spaces. Tensor algebra, symmetry Cartesian Tensors: Pre-requisite MTS 201, 203.
Functions of a complex variable, Limits and continuity of functions of a complex variable. Derivation of the Caychy-Riemann equations. Analytic functions. Bilinear transformations, Conformal mapping, Contour integrals. Cauchy’s theorems and its main Consequences.
Convergence of sequences and series of functions of a complex variable. Power series. Taylor
series. Pre-requisite MTS 206.
Introduction to the language and Concepts of modern mathematics. Basic sets theory mappings, relations, equivalence and other relations, Cartesian products. Binary logic, method of proof. Binary operations. Algebraic structures, semi-groups, rings integral domains fields. Homomorphism. Number systems, properties of integers, rational, real and complex numbers pre-requisite MTS 101.
Sets, matrices and examples. Open spheres (or balls).m Open-sets and neighbourhoods. Closed Sets. Interior, exterior, frontier, limit points and closure of a set. Dense subsets and separable space, Convergence in metric space homoeomorphism. Continuity and compactness, Connectedness. Pre-requisite MTS 209.
Riemann integral of functions. (R…) R, Continuous monopositive functions. Functions of bounded variation. The Riemann strieltjes integra. Pointwise and Uniform Convergence of sequence and series of function R…) R, Effects of limits (sums) when the functions are Continuous differentiable or Riemann integrable power series. Pre-requisite MTS 206.
Methodology of model building: identification, formulation and solution of problems, cause effects diagrams, Equation types. Algebraic, ordinary differential, partial differential, difference, integral and functional equations. Application of mathematical models to Biological, social and behavioural sciences. Pre-requisite MTS 201, 309.
Normal subgroups and quotient groups. Homomorphism isomorphism theorems. Cayley’s theorems. Direct products. Groups of small order. Group acting on sets. Sylow’s theorems. Ideal and quotient rings. P.I.D’s, U.F.D’s Euclidean rings. Irreducibility; field extensions, degree of an extension, minimum polynomial,Algebraic and transcendental extension. Prerequisite MTS 307.
Series solutions of second order linear equations. Bessel Legendre and hypergeometric equations and functions. Gamma, Beta functions Sturm-Liouville problems. Orthogonal polynomials and functions. Fourier, Fourier Bessel and Fourier-Legendre Series. Fourier transfor
mation, Solution of Laplace, wave and heat equations by Fourier method. Pre-requisite MTS 209.
Groups and subgroups, Group Axioms. Permutation Groups, Cosets, Graphs Directed and undirected graphs, subgraphs, cycles connectivity, Application (Flow Charts) and state transition graphs, lattices and Boolean Algebra, finite fields; Minipoly-nomials. Irreducible polynomials, polynomial roots. Application (errorcorrecting codes sequences generators). Pre-requisite MTS 201, 209.
Degree of freedom. Holonomnic and non-holonomic constraints. Generalized co-ordinates language’s equations for holonomic systems. Force dependent on coordinates only, force obtainable from a potential impulsive force.
Co-ordinate in R3. Polar co-ordinates, Distances between points, surfaces and curve in spaces. The plane, straight line. Basic projective Geometry, Affine and Euclidean Geometry’s.
Bivariate normal distribution, the Gama, Chi-square, 2 types of beta, F and T distribution of functions of random variable.; cumulative distribution function, moment generating function and transformation techniques. Probability integral transformation. Order statistics and their functions. Pre-requisite MTS 222.
Brief revision of basic concepts. Probability generating functions. Univariate characteristic functions. Formulas. Various models of convergence. Law of large numbers and the central limit theorem using characteristic functions. Random works and markov chains. Introduction
to poission processes.
Multiple regression and the general linear models. Discrimination and classification, contingency table and non-parametric methods computer-aided design and analysis of experiments, factorial, Taguchi and other designs. Analysis of survival, reliability and quality
control data. Simulation techniques in reliability analysis, uses of s-PLUS, SPSS, MATLAB, STATISTICAL, GENSTAT for data analysis.
Basic principles of experimentation. Randomisation. Replication and local control. Randomised blocks and latin square designs, factorial experiments with factors at two levels. Analysis of experimental data. Confounding and other special problems. Applications in experimental
fields like agriculture, biology and industry. Analysis of variance of complex nested and crossed classification. Incomplete Block designs. Field and laboratory appraisal of some of the techniques and problems in sample surveys.
Basic sampling methods. Stratification. Use of auxiliary information. Multi-state sampling. Non-sampling errors. Estimation of population mean and total in simple and in stratified random sampling. Methods of social investigation; planning surveys, problems, design of surveys, errors and bias, methods of collection of data processing analysis and interpretation. Nigeria’s experience in sampling survey. Pre-requisite MTS 321.
Classical Methods of Optimization Maxima and Minima, Langrange’s Multipliers. Linear programming, convex sets and functions, simplex and revised simplex methods, sensitivity analysis, duality theory, Games Theory, Two person’s zero sum games, saddle point, dominance, strategies.
Floating – point arithmetic, use of mathematical subroutine packages; interpolation; approximation, numerical integration and differentiation; numerical solutions or ordinary differential equations-initial valued problems (IVP).
Lebesque measure; measurable set. Measurable functions Lebesque integral; integration of non-negesative functions, the general integral convergence theorems. Pre-requisite MTS 206, 305.
Vector functions of a real variable. Soundedness. Limits. Continuity and differentiability. Functions of class Cm. Taylor’s formulae. Analytic functions. Cuves: regular, differentiable and smooth. Curvature ad torsion. Tangent line and normal plans. Vector: Functions of Vector variable: Linear continuity and limits. Directional functions of Class Cm. Taylor’s theorem
and inverse function theorem. Concept of a surface;parametric representation, tangent plane and normal lines. Topological properties of simple surfaces. Prerequisite MTS 312.
Existence and uniqueness theorems, Dependence of solution on initial data and parameter. Properties of solutions, Sturm comparison and Soni-polya theorems. Linear systes, Floquet’s theorem. Non-linear Systems; stability theory. Integral equation: classification. Fredholm’s equations; Fredholm’s alternative; method of successive approximations; Neumann’s series; Resolvent Kernel. Volterra equations. Applications to ordinary Differential equations.
General motion of rigid body as a translation plus a rotation. Moment, and products of inertia in three dimensions. Parallel, and perpendicular axes theorems. Principal axes, Angular momentum, Kinetic energy of a rigid body, impulsive motion. Examples involving one and
two dimensional motion of simple systems. Moving frames of reference, rotating and translating frames of reference. Coriolis force. Motion near the Earth’s Surface.
The Foucault’s pendulum. Euler’s dynamical equations for motiomn of a rigid body with one point fixed. The symmetrical top. Procession.
Real and Ideal fluids. Differentiation following the motion of fluid particles. Equations of motion and continuity for incompressible inviscid fluids. Velocity potentials and Stoke’s Stream functions. Bernoulli’s equation with application to flow along curve paths. Kinetic energy. Sources, sinks, doubles in 2- and 3-dimensions, limiting stream lines. Images and rigid planes. Pre-requisite MTS 311.
Process control: Use of control charts to achieve process stability. Tolerance limits as a function of component variability. Product control: design of simple, double multiple, and sequential sampling plans. Cumulative sum charts, feedback theory for controlling continuous process. Pre-requisite MTS 222.
Dynamic and Integral programming, probabilisticmodels. Network Analysis and project control. Transportationand inventory models. Queuing theory.
Point estimation by least squares and maximum likelihood methods. Properties of point estimator; unbiasedness, sufficiency, completeness, uniformly minimum variance unbiasedness. Rao-Cramer inequality, consistence, efficiency, best asymptotic normality.
Confidence intervals and regions. General methods of finding a confidence hound, large sample confidence intervals. Gauss-Markov and Fisher-Cochran theorems Tests of hypothesis: Neymann-pearson theorem. Prerequisite MTS 221.
Purpose, history and structure of biological assays. International standards. Statistical Science and biological assays. Terminology and notations. Types of biological assays. Nature of direct assays. Applications to strophanthus. Precision of estimates.
Calculus of variation: Langrange’s function and associated density. Necessary condition for a week relative extremum. Hamilton’s principles. Lagranges equations and geodesic problems. The Du Bios-Raymond equation and corner conditions. Variable end points and related
theorems. Sufficient conditions for a minimum. Isoperimetric problems. Variational integral transforms. Laplace, Fourier and Hankel transforms. Complex variable methods convolution theorems. Application to solution of differential equations. Pre-requisite MTS 201, 304, 401.
Topological spaces, definition, open and close setsneighbourhoods. Coarse, and finer topologies. Basisand sub-bases. Separatic axioms, compactness, localcompactness, connectedness. Construction of new topologicalspaces from given ones; sub-spaces, quotient
spaces. Continuous functions, homemorphones, topologicalinvariant, spaces of continuous functions:Pointivise and uniform convergence. Pre-requisite MTS 304.
Complete metric space. Functions on metric space, continuity, homeomorphism. Normed Liner Spaces: Convex sets, functions linear operators. Elementary spectral theory for operators in Hilbert space.
Laurent expansions. Isolated singularities and residues. Residue theorem Calculus of residue, and application to evaluation of integrals and to sum of series. Maximum Modulus principle. Argument principle. Rouche’s theorem. The fundamental theorem of algebra. Principle of analytic continuation. Multiple valued functions and Riemann surfaces. Pre-requisite MTS 206, 303.
Lyapunov Theorems. Solution of Lyapunov stability equation ATP+PA = Q. controllability and observability. Theorem on existence of solution of linear systems of differential operations with constant coefficients.
Solution of boundary and eigenvalue problems of PDE by various identified methods. Numerical solution of PDES especially parabolic, and hyperbolic, and hyperbolic systems.
Particles in a gravitational field. curvilinear coordinates, intervals. Covariant differentiation; Christofell symbol and metric tensor. The Constant gravitational field. rotation. The curvature tensor. The action function for the gravitational field. the eneary momentum tensor. Newton’s Law. Motion in a centrally symmetric gravitational field. the energy momentum pseudo-tensor.
Gravitational waves. Gravitational fields at large distances from bodies. Isotropic space. Space-time metric in the closed and in the open isotropic models. The reshift.
Particle gravitational field: Curvilinear coordinates, intervals. Covariant differentiation. Christffel symbol and metric tensor. The constant gravitational field. rotation.
Dynamical Meteorology: The equations of motion, continuity and Thermodynamics is Pressure Coordinates. Pure types of wave motion in the atmosphere. Filtering. Finite difference methods of solution. The equivalent- Barotropic model and Numerical Production. Oceanography: Bussineq, Geostrophic, of plane and plane approximations small amplitude and shallow water approaches. Thin ocean equations and plane Tidal equations.
Navier-Strokes equations for a viscous incompressible fluid. Dynamical similarity and Reynolds number. Steady one-dimensional flow of viscous flow. 2- D flow and small disturbance theory. Radial flow between plane walls, axi-symmetric jets. Boundary layer theory. Inviscid compressible flow – the energy equation and energetics, compressibility effect, unsteady 1-D flow. Equations of motion for some specific types of flow and the ensuring solutions.
Particles-wave duality. Quantum postulates. Schroedinger equation of motion. Potential steps and wells in 1-dim Heisenberg formulation. Classical limit of Quantum mechanics. Computer brackets. Linear harmonic oscillator. Angular momentum. 3-dim square well potential. The hydrogen atom Collision in 3-dim. Approximately methods for stationery problems.
Multivariate normal and related distributions. Inference about mean vectors, Hotellings T2 and Mahalanobis D2 statistics. Discrimination and classification. Tests of independence. Principal components and factor analysis. Pre-requisite MTS 321, 425.
Random walk, simple and general random walk with absorbing and reflecting barriers. Markovian processes with finite chains. Limit theorem. Possion, branching, birth and death processes. Queuing processes; M/M1, M/M/S, M/G/1 queues and their waiting time distributions. Relevant applications. Pre-requisite MTS 322, 425.
Linear programming models. The simplex: Formulationand theory. Quality integer programming; Transportation problem. Two-person zero-sum games. Nonlinear programming; quadratic programming Kuhn-tucker methods. Optimality criteria. Simple variable optimisation. Multivariable techniques. Gradient methods. Pre-requisite MTS 201, 209, 309.
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