Elementary Set Theory: Subsets, Union, Intersection, complements, Venn diagrams. Relation and Function. Real numbers, rational and irrational numbers, mathematical induction, real sequences and series. Theory of quadratic equations. The remainder and factor theorems. Zeros of polynomials, quadratic formula, Binomial theorem. Graphs of polynomial and rational functions, partial fraction decomposition. Linear equations: Linear equations and consistency, Matrices and determinants of second and third order.
Text books
1. Set Theory and Related Topics, S. Lipschutz, (Schaum’s Outline Series), McGraw-Hill (1964).

Trigonometric functions: Radian measure, Sum, difference and product formulas. Trigonometric identities. Inverse trigonometric functions. Solution of trigonometric equations. Graph of trigonometric functions. Half angle formula. Properties and solution of triangles.
Complex numbers: Representation in the plane. Sum, product, Modulus, argument. Complex conjugate and its properties. nth roots. Argand diagram. Demoivre theorem.
Text books
1. Pure Mathematics Books I & II, J.K. Backhouse et al, Longman (1980).
2. Theory and Problems of Trigonometry, Frank Ayres, (Schaum’s Outline Series). (1954).

Geometric representation of vectors in 1-3 dimension components, direction cosines. Addition, Scalar, multiplication of vectors, linear independence. Scalar and vector product of two vectors. Two dimensional coordinate geometry. Straight lines, circles, parabola, ellipse, hyperbola. Tangents, normal Impact of two smooth spheres and of a sphere on smooth sphere.
Text books
1. Calculus and Analytic Geometry, G.B. Thomas and R.L.Finney, Addison-Wesley (1979).
2. Vector Analysis, Murray R. Spiegel, Schaum’s Outline Series (1974)

Functions: Domain and range. Inverse function. Concept of limits. Standard limits. Techniques of finding limits. Continuity. Continuous functions and their properties. Differentiation: Derivatives. Geometric interpretations. Rates of change and derivatives. Differentiation techniques. Derivative of elementary functions. Second order derivative. Applications of differentiation in geometry, curve sketching, maxima and minima.
Integration: The definite integral. Integration techniques. Indefinite integral. Computation and properties. Definite integral as the limit of a sum. Application in areas, volumes and centre of mass. Integration of rational functions, trigonometric integral, trigonometric substitutions, Trapezium rule.
Text books
1. Calculus and Analytic Geometry, G.B. Thomas and R.L.Finney, Addison-Wesley (1979).

Particles, rigid bodies. Frame of reference. Units’ displacement, velocity and acceleration. Force. Fundamental laws of Newtonian mechanics. The theory of dimension and dimensional analysis. Statics: Equilibrium of a particle. Equilibrium of a system of particles. Mass canter and Centre of gravity. Friction. Kinematics: One and two dimensional motions. Circular motions. Relative motions. Dynamics: laws of inertia and linear momentum. Friction and elastic force. Motion of planets. Impulses, work and energy.
Text books.
1. Textbook of Dynamics, F. Charlton, Ellis Horwood, 1977.
2. Future mathematics project Tuttuh-Adegun NPS Educational. (1991).

History of Computers; functional components of computer; characteristics of a computer; Hardware and software; problem solving; flow charts, Algorithms; Computer languages, programming and packages, statements symbolic names; arrays, subscripts, expressions and control statements. Introduction to BASIC Programming Language, computer application.

Vector space over the real field. Subspaces, linear independence, basis and dimension. Linear transformations and their representation by matrices – range, null space, rank. Singular and non-singular transformation and matrices. Algebra of matrices.
Text books
1. Linear Algebra, S. lipschutz (Schaum’s Outline Series) Mc Graw-Hill (1987).
2. Linear Algebra and Matrix Theory, E.D. Nerring, John Willey, (1967).

Real-valued functions of a real variable. Review of differentiation and integration and their applications. Mean value theorem. Taylor series. Real-valued functions of two or three variables. Partial derivatives chain rule, extreme, languages multipliers. Increments, differentials and linear approximations. Evaluations of line, integrals. Multiple integrals.
Text books
1. Calculus and Analytic Geometry, G.B. Thomas and R.L.Finney, Addison-Wesley (1979).

Bounds of real numbers, convergence of sequence of numbers. Monolone sequences, the theorem of nested Intervals. Cauchy sequences, tests for convergence of series. Absolute and conditional convergence of series and rearrangements. Completeness of reals and incompleteness of rationales. Continuity/and differentiability of functions R….) R. Rolles’s and mean value theorems for differentiable functions Taylor series.
Text books
1. Real and complex analysis, Walter Rudin, Tata Mc Graw-Hill (1974).
2. Complex Variable, J.N.Sharma, Pragti Publication India.

Systems of linear equation, change of basis, equivalence and similarities. Eigenvalues and eigenvectors. Minimum and characteristics polynomials of a linear transformation (Matrix). Caley-Hamilton. Bilinear and quadratic forms, orthogonal diagnolisation. Canonical forms.
Text books
1. Linear Algebra, S. lipschutz (Schaum’s Outline Series) Mc Graw-Hill (1987).
2. Linear Algebra and Matrix Theory, E.D. Nerring, John Willey, (1967).
3. Matrix JN Kapur New Age International (1998).

Solution of algebraic and transcendental equations. Curve fitting. Error analysis. Interpolation and approximation. Zeros or non-linear equations ‘to one variable’. Systems of linear equations. Numerical differentiation and integral equations. Initial value problems for ordinary differential equation.
Text books
1. Numerical methods, Santosh Kumar Gupta, New Age International publisher.
2. Numerical Methods, B.S.Grewal. Khanna Publisher.

Introduction to the language and concepts of modern Mathematics. Topics include; Basic set theory: mappings, relations, equivalence and other relations, Cartesian products. Binary logic, methods of proof. Binary operations. Algebraic structures, semi groups, rings, integral domains fields. Homeomaphics. Number systems; properties of integers, rational, real and complex numbers.
Text books
1. Fundamentals of mathematical analysis, G.Das, S.Patnayak, Tata Mc Grawhill publisher.
2. Discrete mathematics, CLiu, Tata MCGraw hill. Publisher.

Derivation of differential equations from primitive, geometry, physics etc. order and degree of differential equation. Techniques for solving first and second order linear and non-linear equations. Solutions of systems of first order linear equations. Finite linear difference equations. Application to geometry and physics.
Text books
1. Theory and Problems of Advanced Mathematics for Engineers and Scientists, M.R. Spiegel (Schaum’s Outline Series) Mc Graw-Hill (1974).
2. Theory and Problems of Differential Equations, Frank Ayres Jr. (Schaum’s Outline Series). McGraw-Hill, (1972).
3. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).

Elementary vector algebra, Vector and Tripple vector products (more application solution of vector equation, plain curves and space curves. Geometrical equation of lines and planes. Linear independence of vectors; components of vectors, direction cosines; position vector and scalar products; senent frenent formulae; differential definition of gradients, divergent and simple multiplication)
Text books
1. Vector Analysis, Murray R. Spiegel, Schaum’s Outline Series (1974)

Riemann integral of functions R, continuous monopositive functions. Functions of bounded variation. The Riemann Strieltjes integral. Pointwise and uniform convergence of sequences and series of functions R….)R. Effects on limits (sums) when the functions are continuous differentiable or Riemann integrable power series.
Text books
1. Real and complex analysis, Walter Rudin, Tata Mc Graw-Hill (1974).

Methodology of model building; Identification, formulation and solution of problems, cause-effect diagrams Equation types. Algebraic, ordinary differential, partial differential, difference, integral and functional equations. Application of mathematical models to pluprical, biological, social and behavioral sciences.
Text books
1. Mathematical modeling J.N.Kapur New Age Publisher New Delhi.

Functions of a complex variable. Limits and continuity of functions of a complex variable. Derivating the Cauchy-Riemann equations. Analytic functions. Bilinear transformations, conformal mappling Contour integrals. Cauchy’s theorems and its main consequences, Convergence of sequences and series of functions of a complex variable. Power series. Taylor series
Text books
1. Mathematical Methods, J. Heading, University Press, (1963).
2. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).

Group: definition, examples including permutation grounds. Subgroups, cosets. Legranges theorem and applications. Cyclic, groups. Rings: definition examples including Z., Zn, rings of polynomials and matrices. Integral domains, fields, polynomial rings, factorization. Euclidean algorithm for polynomials H.C.F. and L.C.M. of polynomials.
Text books
1. Algebra for College Students, Mark Dugoplki Tata Mc Graw hill. Publisher.
2. Abstract Allgebra, R.M.Khan New central book Agency Ltd.

Polynomial and splines approximation. Orthogonal polynomials and chebysev approximations. Direct and interactive methods for the solution of systems of linear equations. Eigen value problem – power methods, inverse power methods. Pivoting strategies.
Text books
1. Mathematical Methods, J. Heading, University Press, (1963).
2. Numerical Methods. Jain, Jain& Iyenger New Age International. Publisher.

Ordinary differential equations: linear dependence, Wronskian, reduction order, variation of parameters, series solution about ordinary and regular points. Special functions: Gamma, Beta, Bessel, lengendre, Hyper geometric. Laplace transform and application to initial value problems.
Text books
1. Mathematical Methods, Afken & Weber, Elsevier Academic Press.
2. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).

Linear programming models. The simplex Method: formulation and theory. Quality integer programming; Transportation problem. Two-person zero-sum games. Nonlinear programming: quadratic programming Kuhn-tucker methods. Optimality criteria. Simple variable optimization. Multivariable techniques. Gradient methods.
Text books
1. Operational Research PK Gupta, D.S.Hira S Chand Publication Delhi.
2. Operational Research C.K Mustaffi. New Age Publication.

Normal subgroups and quotient groups. Monomorphic isomorphism theorems. Cayley’s theorem. Direct products. Groups of small order. Group acting on sets. Sylow theorems. Ideal and quotient rings. P.I.D. 8, U.F.D’s euclides rings. Irreducibility; Field extensions, degree of an extension, minimum polynomial. Algebraic and transcendental extensions. Straight edged and compass constructions.
Text books
1. Abstuct Algebra . R.M Khan ,Central Book Agency Ltd India.
2. Fundamental of Abstract Algebra. Malik ,M.K .Sen Tata MC.gwawal hill PublisherI.

Sets, matrices, and examples. Open spheres (or balls). Open sets and neighborhoods. Closed sets. Interior, exterior, frontier, limit points and closure of a set. Dense subsets and separable space. Convergence in metric space homeomorphisms. Continuity and compactness, connectedness.
Text books
1. Real and complex analysis, Walter Rudin, Tata Mc Graw-Hill (1974).
2. Introduction to Functional Analysis, Taylor, A. E. John Wiley (1958)

Sturm – Liouville problem. Orthogonal polynomials and functions. Fourier series and integrals. Partial differential equations: general and particular solutions. Linear equations with constant coefficients, first and second order equations, solutions of the heat, wave and Laplace equations by the method of separation of variables. Eigen on of parameters. Fourier transforms.
Text books
1. Mathematical Methods, J. Heading, University Press, (1963).
2. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).
3. Mathematical methods, Gupta& Malik Pragti Publication India.
4. Mathematical Methods, J. Heading, University Press (1963).

Laurent expansions. Isolated singularities and residues. Residue theorem Calculus of residue, and application to evaluation of integrals and to summate of series. Maximum Modulus principle. Arguent principle. Rouche’s theorem. The fundamental theorem of algebra. Principle of analytic continuation. Multiple valued functions and Riemann surfaces.
Text books
1. Mathematical Methods, J. Heading, University Press, (1963).
2. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).

Graph theory. Definition of graph, and diagraph. Complete graphs. Bipartite graphs, planner graphs. Subgraphs. Walks,paths, cycles, connetedness. Applications. Shortest path problem. Trees, spanning trees, the connector problem. Eueran and Hamiltorian graphs. Coloring Reachabilty in a digraph. Network flows. Generating functions, recurrence relations.
Text books
1. Elements of Discrete mathematics, CLiu , D Mahapatra , Tata Mc. Graw hill. Publisher.
2. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).
3. Graphy Theory, Narsing Deo, Prentice Hall India Ltd.
4. Graph Theory, Prabhakar Gupta, Pragti Publication Meerut India.

Vector algebra. Vector, dot and cross Products. Equating of curves and surfaces. Vector differentiation and applications. Gradient, divergence and curl. Vector integrate, line surface and volume integrals Green Stoke’s and divergence theorems. Tensor products of vector spaces. Tensor algebra. Symmetry. Cartesian tensors.
Text books Vector Analysis, Murray R. Spiegel, Schaum’s Outline Series (1974).

Lebesgue measure; measurable and non-measurable sets. Measurable functions. Lebesgue integral: Integration of non-negative functions, the general integral, convergence theorems.
Text books
1. Measure Theory and Integration G.Debrra, New Age Publisher.
2. Measure Theory J. N. Sharma, Krishna Publication India.

Topological spaces, definition, open and closed sets, neighborhoods. Coarser, and finer topologies. Basis and sub-bases. Separable axioms, compactness, local compactness, connectedness. Construction of new topological spaces from given ones; Sub-spaces, quotient spaces. Continuous functions, homeomorphism, topological invariants, spaces of continuous functions: Point vise and uniform convergence.
Text books
1. Topology General Algebra, D.Charjee New Age International Publication.
2. Introduction to General Topology, K.D. Joshi New Age International Publication.

Linear programming: Simplex method. Dual simplex method. Postoptimaity sensitivity analysis. Parametric linear programming. Transportation of problem. Nonlinear programming: Classing optimization theory, Unconstrained and Integer linear programming, and Integer nonlinear programming.
Text books
1. Operations Research PK. Gupta, D.S. Hira. S.Chand Publisher.
2. Operations Research, Mustaffi. New Age International Publisher.

Differential equations: existence and uniqueness theorems dependence of solution on initial data and parameters. Properties of Solutions. Sturm comparison and sonin-Polya theorems. Linear and non-linear systems. Floguet’s theory and stability theory. Integral equations: classification, volterra and fredlhom types. Neumann series. Fredholm alternative for degenerate Hilbert – Schmidt kernels. Reduction of ordinary differential equations to integral equations, Riemann of ordinary differential equations to integral equations. Systematic kernels, eigen function expansion with application.
Text books
1. Theory and Problems of Advanced Mathematics for Engineers and Scientists, M.R. Spiegel (Schaum’s Outline Series) Mc Graw-Hill (1974).
2. Theory and Problems of Differential Equations, Frank Ayres Jr. (Schaum’s Outline Series). McGraw-Hill, (1972).
3. Advanced Engineering Mathematics, E. Kreyszig, Willey, (1987).
4. Integral Equation, R.K Gupta, Shanti Swurup . Krisha Publication India.

Some basic concepts. Further Group Actions. Series, character subgroups, and commutators. P-Groups and Application of Sylow Theorems. Finite Nilpotent and soluble groups. Transitive and primitive Groups.
Text books
1. Rings, Field and Groups (Introduction to Abstract Algebra), R.B.J.T. Allenby, Edward Arnold, (1993).
2. Abstract Algebra, R.M Khan. Cental Book Agency Landon.

Finite difference equation and operations; Discrete variable methods for solution of IUPS – ODES. Discrete and continuous Tan methods for solving IUP – ODES, error analysis. Partial differential equation. Finite difference and finite elements methods. Stability convergence and error analysis.
Text books
1. Numerical methods, Santosh Kumar Gupta, New Age International publisher.
2. Numerical methods for engineers and scientist, New Age International Publisher India.
3. Numerical Methods, B. S. Grewal. Khanna Publisher India.

Project Scheduling by PERM-COM. Network representations. Stochastic Programming: Stochastic linear programming. Stochastic nonlinear Programming. Stochastic dynamic programming. Dynamic programming: Decision theory and Games: Decisions under risk. Decisions under uncertainty. Game theory. Inventory Models: Deterministic and probabilistic models.
Text books
1. Operations Research PK. Gupta, D.S. Hira. S.Chand Publisher.
2. Operations Research, C.K. Mustaffi. New Age International Publisher.

Hilbert Spaces, bounded linear functional, operators and Banach spaces, topological vector spaces, Banach algebra.
Text books
1. Functional Analysis B.V.Lemey Tata Mc Graw hill.
2. Functional Analysis, J.N.Sharma, A.R Vashistha, Krishna Publication India.

Calculus of variation: Lagrange’s functional and associated density. Necessary condition for a week relative extremum. Hamilton’s principles. Lagrange’s equations and geodesic problems. The Du Bois-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.

Congruences. Chinese remainder theorem. Linear diophntine equations. Quadratic residues. Law of quadratic reciprocity. Continued fractions. Quadratic irrationality. Bell’s equations. Sums of two and four squares. Arithmetic functions and the greases integer function. The Mobius function. Algebraic numbers and algebraic integers. Integers in quadratic fields. Fermat’s last theorem.
Text books
1. Elements of Number Theory, Devid Burten Tata Mc Graw hill publisher.

Theory and solutions of first-order and second order linear equations. Classifications, characteristics, canonical forms, Cauchy problems. Elliptic equations; Laplace’s and Poisson’s formulas, properties of harmonic functions. Hyperbolic equations; wave equations, retarded potential; transmission line equation, Riemann method. Parabolic equation. Diffusion equation, singularity function, boundary and initial – value problem.
Text books
1. Mathematical Methods for physicits Arfken& Weber ,Elsevier Academic Press.

Representations, reducibility. Induced representations. Characters Schur’s Lemma. Orthogonality of characters. Representations of symmetric groups, Symmetric groups. Representations of p-groups and p-groups.
Text books
1. Rings, Field and Groups (Introduction to Abstract Algebra), R.B.J.T. Allenby, Edward Arnold, (1993).
2. Abstract Algebra, R.M Khan. Cental Book Agency Landon.
3. Elements of Number Theory, Devid Burten Tata Mc Graw hill publisher.

Vector functions of a real variable. Handedness. Limits. Continuity and differentiability. Functions Cm. Taylor’s Formulae. Analytic functions. Curves: regular, differentiable and smooth. Curvature and 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.
Text books
1. A course in Differential Geometry, Wilhelm Klingenberg, Springer-Verlag. 1978.
2. Differential Geometry, Loius Auslander, Harper’s Series, 1967
3. Differential Geometry of three dimensions, C.E. Weatherburn, Cambridge University Press, 1964.

This course give students the opportunity to study in-depth a topic in mathematics or statistics that particularly interest them. This may broaden the student’s intellectual horizon through research orientation. This may include reading mathematics or statistical work using advanced method to solve a given problem and explore new areas etc.