Maths

MTE-01

CALCULUS

4 Credits

Basic properties of R, Absolute value, Intervals on the real line, Functions (Definition and examples), Inverse functions, Graphs of functions, Operations on functions, Composite of functions, Even and odd functions, Monotone functions, Periodic functions. Definition of limits, Algebra of limits, Limits as x → ∞ (or − ∞ ), One-sided limits, Continuity (Definitions and Examples, Algebra of continuous functions), Definition of derivative of a function, Derivatives of some simple functions, Algebra of derivatives, The chain rule, Continuity versus derivability. Derivatives of the various trigonometric functions, Derivatives of inverse function, The inverse function theorem, Derivatives of inverse trigonometric functions, Use of transformations. Derivative of exponential function, Logarithmic functions, Hyperbolic functions, Inverse hyperbolic functions, Methods of differentiation (Derivative of xr, Logarithmic Differentiation, Derivatives of functions defined in terms of a parameter, Derivatives of implicit functions). Second and third order derivatives, nth order derivatives, Leibniz theorem, Taylor’s series and Maclaurin’s series Maxima-minima of functions (Definitions and examples, a necessary condition for the existence of extreme points), Mean value theorems (Rolle’s theorem, Lagrange’s mean value theorem), Sufficient conditions for the existence of extreme points (First derivative test, Second derivative test), Concavity/convexity, Points of inflection. Equation of tangents and normals, Angles of intersection of two curves, Tangents at the origin, Classifying singular points, Asymptotes (Parallel to the axes, Oblique asymptotes). Graphing a function, Tracing a curve (given its Cartesian equation, or in parametric form, or Polar equation). Partitions of a closed interval, Upper and lower product sums, Upper and lower integrals, Definite integral, Fundamental theorem of calculus. Standard integrals, Algebra of integrals, Integration by substitution, Integrals using trigonometric formulas, Trigonometric and Hyperbolic substitutions, Two properties of definite 2 2 2 2 2 2 integrals, Integration by parts, Evaluation of ∫ (a − x )dx, ∫ (a + x )dx, ∫ ( x − a )dx,

∫ e (f (x) + f ′(x ))dx . Reduction formulas for ∫ sin x dx, ∫ cos x dx, ∫ tan x dx and ∫ sec x dx, Integrals involving products of trigonometric functions (Integrand x

n

n

n

n

of the type sin m x cos n x, e ax sin n x ), Integrals involving hyperbolic functions. Integration of some simple rational functions, Partial fraction decomposition, Method of substitution, Integration of rational trigonometric functions, Integration of Irrational functions Monotonic functions, Inequalities, Approximate value. Area under a curve (Cartesian equation, Polar equations), Area bounded by a closed curve, Numerical integration. (Trapezoidal rule, Simpson’s rule). Length of a plane curve (Cartesian form, Parametric form, Polar form), Volume of a solid of revolution, Area of surface of revolution. Video Programme: Curves

MTE-02

LINEAR ALGEBRA

4 Credits

Sets, subsets, union and intersection of sets, Venn diagrams, Cartesian product, relations, functions, composition of functions, binary operations, fields. Plane and space vectors, addition and scalar multiplication of vectors, scalar product, orthonormal basis, vector equations of a line, plane and sphere. Definition and basic properties, subspaces, linear combination, algebra of subspaces, quotient spaces. Linear independence and some results about it, basic results about basis and dimension, completion of a linearly independent set to a basis, dimension of subspaces and quotient spaces. Definitions and examples of linear transformation, kernel, range space, rank and nullity, homomorphism theorems. L (U, V), the dual space, composition of transformations, the minimal polynomial. Definition of a matrix, matrix associated to a linear transformation, the vector space Mmxn(F), transpose, conjugate, diagonal and triangular matrices, matrix multiplication, inverse of a matrix, matrix of a change of basis. Rank of a matrix, elementary operations, row-reduced echelon matrices, applying row reduction to obtain the inverse of a matrix and for solving a system of linear equations. Definition and properties, product formula, matrix adjoint and its use for obtaining inverses, Cramer’s rule, determinant rank. Definition and how to obtain them, diagonalisation. Cayley-Hamilton theorem, minimal polynomial’s properties. Definition, norm of a vector, orthogonality. Linear functionals of inner product spaces, adjoint of an operator, self-adjoint and unitary operators, Hermitian and unitary matrices. Definitions, representation as matrix product, transformation under change of basis, rank of a form, orthogonal and normal canonical reductions Definitions, standard equations, description and some geometrical properties of an ellipse, a hyperbola and parabola, the general reduction. Video Programme: Linear Transformations and Matrices

MTE-03

MATHEMATICAL METHODS

4 Credits

Sets, Equality of Sets, Operations on Sets, Venn diagrams, Functions, Types of Functions, Composite Functions, Operations with Functions. Graphs (Exponential and Logarithmic Functions, Trigonometric functions), Trigonometric ratios. Polynomials and Equations, Sequences and Series, Permutations and Combinations, Binomial Theorem. Two Dimensional Coordinate System -Distance between Two points, Area of a Triangle, Equation of a Line, , Angle between Two Lines, Distance of a Point from a Line, Circle, Three Dimensional Coordinate System- Equation of a Straight Line in 3-D, The Plane, The Sphere. Vectors as directed line segments, Algebra of Vectors and their applications (addition and subtraction of vectors, resolution of vectors, dot and cross product). Limit and Continuity, Derivative of a Function at a Point, Its Geometrical Significance, Rules for differentiation, differentiation of Trigonometric, Exponential and Logarithmic Functions, Differentiation of Inverse Algebraic and Inverse Trigonometric Functions, Chain Rule, Differentiation of Implicit Functions and Logarithmic Differentiation, Physical Aspects of Derivatives Tangents and Normals, Higher Order Derivatives, Maxima and Minima, Asymptotes, Curve-Tracing, Functions of Two Variables, Partial Derivatives of Order Two, Homogeneous Functions, Euler’s Theorem. Antiderivatives, Integration as Inverse of Differentiation, Definite Integral as the Limit of the Sum, Properties of Definite Integrals, Fundamental Theorem of Integral Calculus Standard Integrals, Methods of Integration, Integration by Substitution, Integration by parts, Integration of Trigonometric Functions Preliminaries, Formation of Differential Equations, Methods of Solving Differential Equations of First Order and First degree (Variables Separable, Homogeneous Equation, Exact Equations and Linear Equations). Some Basic Definitions in Statistics, Frequency Distribution, Discrete Random Variables, Continuous Random Variables, Measures of Central Tendency and Dispersion (Mean, Mode, Median, Standard Deviation, Mean Deviation). Preliminaries: (Sample Space, Discrete Sample Space, Continuous Sample Space), Rules of Probability, Conditional Probability, Baye’s Theorem. Combination of events, Binomial Distribution, Poisson Distribution (Emphasis Through Illustrations). Continuous Random Variables, Types of Continuous Distributions (Exponential and Normal Distribution – Emphasis Through Illustrations) Sample Selection, Random Sampling Procedure, Measure of Variation and Accuracy, Standard Error, Unbiased Estimator, Accuracy and Precision of Sample Estimator, Types of Sample Design (Random Sampling, Cluster Sampling).Statistical Hypothesis, Level of Significance, Degrees of Freedom, Chisquare Test, t-test, Analysis of Variance Correlation and scatter diagram, Correlation coefficient, Linear regression, Curve Fitting (Least Square Method). Video Programme: 1) 2)

Sampling a case study Sampling in Life Sciences

MTE-04

ELEMENTARY ALGEBRA

2 Credits

Definition and examples of sets and subsets, Venn diagrams, Complementation, Intersection, Union, Distributive laws, De Morgan’s laws, Cartesian product. What a complex number is, Geometrical representation, Algebraic operations, De Moivre’s theorem, Trigonometric identities, Roots of a complex number. Recall of solutions of linear & quadratic equations, Cubic equations (Cardano’s solution, Roots and their relation with coefficients), Biquadratic equations (Ferrari’s solution, Descartes’ solution, Roots and their relation with coefficients) APPENDIX: Some mathematical symbols (Implication, two-way implication, for all, their exists), Some methods of proof (Direct proof, contrapositive proof, proof by contradiction, proof by counter-example). Linear systems, Solving by substitution, Solving by elimination. Definition of a matrix, Determinants, Cramer’s rule. Inequalities known to the ancients (Inequality of the means, Triangle inequality), Less ancient inequalities (Cauchy-Schwarz inequality, Weierstrass’ inequalities, Tchebyshev’s inequalities)

MTE-05

ANALYTICAL GEOMETRY

2 Credits

Equations of a line, Symmetry, Change of axes (Translating the axes, rotating the axes), Polar coordinates. Focus-directrix property, Description of standard form of parabola, ellipse and hyperbola; Tangents and normals of parabola, ellipse, hyperbola; Polar equation of conics. General second degree equation, Central and non-central conics, tracing a conic (Central conics, Parabola), Tangents, Intersection of conics. Points, Lines (Direction cosines, Equations of a straight line, Angle between two lines), Planes (Equations of a plane, Intersecting planes and lines). Equations of a sphere, Tangent lines and planes, Two intersecting spheres, Spheres through a given circle. Cones, Tangent plane to a cone, Cylinders. Definition of a conicoid, Change of axes (Translation of axes, projection, Rotation of Axes), Reduction to standard form. A conicoid’s centre, Classification of central conicoids, Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets, Intersection with a line or a plane.Standard equation, Tracing the paraboloids, Intersection with a line or a plane.

MTE-06

ABSTRACT ALGEBRA

4 Credits

Sets, Cartesian Product, Relations, Functions, Some number theory – Principle of induction and divisibility in Z. Binary operations, Definition of a group, Properties of a Group, Some details of Zn, Sn, C, and appendix on some properties of complex numbers. Subgroups and their properties, Cyclic groups. Cosets; Statement, proof and applications of Lagrange’s theorem. Definition and standard properties of normal subgroups, Quotient groups. Definition and examples, Isomorphisms, Isomorphism theorems, Automorphisms. Definition, Examples, Cayley’s theorem. Direct product, Sylow theorems (without proof), Classifying groups of order 1 to 10. Elementary properties, Examples of commutative and non-commutative rings and rings with and without identity. Definitions, Examples, Standard properties, Quotient Rings (in the context of commutative rings). Definition and properties of integral domains, Fields, Prime and maximal ideals, Fields of quotients. Examples, Division Algorithm and Roots of Polynomials. Euclidean domain, PID, UFD. Eisenstein’s criterion, Prime fields, Finite fields Video Programme: Groups of Symmetries

MTE-07

ADVANCED CALCULUS

4 Credit

The Extended Real Number System R ∞ (Arithmetic Operations in R ∞ , Bounds in R ∞ . Extension of Exponential and Logarithmic Functions to R ∞ ). The concept of infinite limits (infinite limits as the independent variable x → a ∈ R , One-sided Infinite Limits, Limits as the independent variable tends to ∞ or - ∞ , Algebra of limits). Indeterminate Forms, L’Hopital’s rule for Rule, Another form of L’Hopital’s rule for

0 0 0 0

form (Simplest form of L’Hopital’s form), L’Hoptal’s rule for

∞ ∞

form,

other types of Indeterminate Forms (indeterminate forms of the type ∞ - ∞ , indeterminate forms of the type 0. ∞ , indeterminate forms of the type 00, ∞ 0,1 ∞ ). The Space Rn (Cartesian products, algebraic structure of Rn, Distance in Rn), Functions from Rn to Rm. Limits of Real-Valued Functions, Continuity of Real-Valued Functions, Limit and Continuity of Functions from Rn → Rm, Repeated limits. First Order Partial Derivatives (Definition and Examples, Geometric interpretation. Continuity and Partial Derivatives), Differentiability of Functions from R2 to R, Differentiability of functions from Rn → R, n > 2. Higher Order Partial Derivatives, Equality of Mixed Partial Derivatives. Chain Rule, Homogeneous Functions, Directional Derivatives Taylor’s Theorem (Taylor’s theorem for functions of one variables, Taylor’s theorem for functions of two variables), Maxima and Minima (local extrema, Second derivative test for local extrema), Lagrange’s multipliers; Jacobians (Definition and examples, Partial derivatives of Implicit Functions), Chain rule, Functional Dependence (Domains in Rn, Dependence). Implicit Function Theorem (Implicit Function Theorem for two variables, implicit Function Theorem for three variables), inverse Function Theorem. Double Integral over a Rectangle (Preliminaries, Double Integrals and Repeated Integrals), Double Integral over any Bounded set (Regions of Type I and Type II, Repeated integrals over regions of Type I and Type II), Change of variables.Integral over a region in space (Integral over a Rectangular Box, Integral over Regions of Type I and Type II), Change of Variables in Triple Integrals (Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates). Applications of double integrals (area of a planar region and volume of a solid, Surface area, Mass and moments), Applications of triple integrals; Line Integrals, Independence of path, Green’s Theorem. Video Programme: Double Integration

MTE-08

DIFFERENTIAL EQUATIONS

4 Credits

Basic concepts in the theory of differential equations, Family of curves and differential equations, Differential Equations arising from physical situations. Separation of Variables, Homogeneous equations, Exact equations, Integrating factors. Classification of first order differential equations (DE), General solutions of linear non-homogeneous equation, Method of Undetermind coefficient, Method of Variation of Parameters, Equations reducible to linear form, Applications of linear DEs. Equations which can be factorized, Equations which cannot be factorized (Solvable for x, y, independent variable absent, homogeneous in x and y, Clairaut’s and Riccati’s equations). General form of linear ordinary differential equation, Condition for the existence of unique solution, linear dependence and independence of the solution of DEs, Method of solving homogeneous equation with constant coefficients. Types of nonhomogeneous terms for which the method is applicable (polynomial, exponential, sinusoidal etc.), Observations and Constraints of the method. Variation of parameters, Reduction of order, Euler’s equations. Differential operators, General method of finding Particular Integral (PI), Short method of finding PI, Euler's equations. Method of changing independent Variable, Method of changing dependent variable, Applications – Mechanical Vibrations, Electric Circuits Frames of reference for curves and surfaces, Basic concepts in 2-dimensions, Curves and surfaces in space. Formation of simultaneous DEs, Existence and Uniqueness, Methods of solution of dx P

=

dy Q

=

dz R

,

Applications – Orthogonal trajectories, Particle motion in phase-space,

Electric Circuits. Formation of pfaffian Differential Equations their geometrical meaning, Integrability, Methods of Integration (Variable separable, One variable separable, Homogeneous PfDEs, Natani’s method). Origin, Classification and Solution of linear first order PDEs, Linear Equations of the First Order, Cauchy Problem. Complete integral, Compatibility, Charpits method, Standard forms, Jacobi’s method, Cauchy problem. General form of partial differential equation of any order, – Classification and the Integral, Solutions of reducible homogeneous equations, Solutions of irreducible homogeneous equations. Particular integral, Analogies of Euler’s equations Origin of second order PDEs, Classification, Variable separable solution for Heat flow, Wave and Laplace equations. Video Programme: Let’s Apply Differential Equations

MTE-09

REAL ANALYSIS

4 Credits

Sets and functions, system of real numbers, mathematical induction. Order relations in real numbers, algebraic structure (ordered field, complete ordered field), countability. Neighbourhood of a point, open sets, limit point of a set (Bulzano-Weiertrass Theorem), closed sets, compact sets (Heine-Borel Theorem, without proof). Algebraic functions, transcendental functions, some special functions. Sequences, bounded sequences, monotonic sequences, convergent sequences, criteria for the convergence of sequences, cauchy sequences, algebra of convergent sequences. Infinite series, general tests of convergence, some special tests of convergence (D’Alembert’s ratio test, Cauchy’s integral test, Raabe’s test, Gauss’s test). Alternating series (Leitnitz’s test), absolute and conditional convergence, rearrangement of series. Notion of limit (finite limits, Infinite limits, sequential limits), algebra of limits. Continuous functions, algebra of continuous functions, non-continuous functions. Continuity on bounded closed intervals, pointwise continuity and uniform continuity. Derivative of a function (geometrical interpretation), differentiability and continuity, algebra of derivatives, sign of a derivatives. Rolle’s theorem, mean value theorems (Lagrange, Cauchy and generalised mean value theorems), intermediate value theorem for derivatives (Darboux theorems). Taylor’s theorem, Maclaurin’s expansion, indeterminate forms, extreme values. Riemann integrability, Riemann integrable functions, Algebra of integrable functions, computing an integral. Properties of Riemann integral, Fundamental Theorem of Calculus, mean value theorems. Sequences of functions, Pointwise convergence, uniform convergence (Cauchy’s criterion), series of functions. Video Programme: Limits – glance through history.

MTE-10

NUMERICAL ANALYSIS

4 Credits

Three fundamental theorems of calculus (Intermediate value theorem, Rolle’s theorem, Lagrange’s mean value theorem) Taylors theorem, Errors (round-off and truncation) Initial approximation to a root (tabulation method, Graphical method) Bisection method, Fixed point Iteration method. Regula-Falsi method, Newton-Raphson method, Convergence criterion. Some Results on roots of polynomial equation, Birge-Vieta method, Graeffe’s root squaring method Direct methods of solving linear algebraic equations-Cramer’s rule, Direct methods for special matrices, Gauss elimination method, LU decomposition method. Finding inverse of a square matrix - The method of adjoints, The Gauss-Jordan reduction method, LU decomposition method. Iterative methods of finding solutions - The Jacobi iteration method, The Gauss-Seidal Iteration method. The eigenvalue problem,The power method, The inverse power method. Lagrange’s form of interpolation, Inverse Interpolation, General error term. Divided differences, Newton’s general form of interpolating polynomials - Error term, Divided difference and derivatives, Further results on Interpolation error. Backward Interpolation at equally spaced points - Forward and Central Differences, Newtons forward difference formula, Newton’s backward difference formula, Strirling’s central difference formula Numerical differentiation - Methods based on undetermine coefficients, On finite difference Operators and on Interpolation, Richardson’s Extrapolation, Optimum Choice of Step Length Numerical Integration - Methods based on Interpolation (Lagrange interpolation, Newton’s Forward Interpolation) Composite Integration, Romberg Integration Numerical solutions of ODE - Basic concepts, Taylor series method, Euler’s method, Richardson’s extrapolation Runge-Kutta methods of second, third and fourth order, Richardson’s Extrapolation.

MTE-11

PROBABILITY AND STATISTICS

4 Credits

Raw materials of statistics, Frequency distributions: Ungroped frequency distributions, Grouped frequency distributions); Diagrammatic representation of frequency distributions: frequencies, Commulative frequencies, Frequency curve, Broad classes of distributions. Central tendency and dispersion, Measures of central tendency; The mean, The median, The mode, Algebraic properties of the measures, A comparison of the measures; Measures of dispersion: The range, The mean deviation, The standard deviation, Algebraic properties of the measures, A comparison of the measures; Coefficient of variation. Moments and quantiles: Moments of a frequency distribution, Quantiles of a frequency distribution; Skewness; Kurtosis. Tabular and diagrammatic representation of Bivariate data; What do we mean by regression analysis?, Simple regression line; Correlation coefficient, Relationship between regression and correlation coefficients, Limitations of correlation coefficient. Random experiment, Sample space, Events, Algebra of events. Probability: Axiomatic Approach: Probability of an event: Definition, Probability of an event: Properties; classical definition of probability, Conditional probability, Independence of events, Repeated experiments and trials, Random variable, Two or more random variables: Joint distribution of random variables, marginal distributions and independence, Mathematical expectation, Variance, Covariance and correlation coefficient, Moments and moment generating function, Distribution of sum of two random variables. The Bernoulli distribution, The binomial distribution, The multinomial distribution, The hypergeometric distribution. The negative binomial distribution, The Poisson distribution. Distribution functions, Density functions, Expectation and variance, Moments and moment generating functions, Functions of a random variable. Uniform distribution, Normal distribution, Exponential and gamma distributions, Beta distribution. Bivariate distributions, Conditional distributions, Independence, Expectation of functions of a random vector, Correlation coefficient, regression. Functions of two random variables: Direct Approach and Transformation approach, Functions of more than two random variables, Chi-square distribution, t-distribution, Fdistribution. Chebyshev’s inequality and weak law of large numbers, Poisson approximation to binomial, Central limit theorem: Normal approximation to binomial. Inductive inference, Random sampling, Sampling distributions related to normal distribution, Point estimation, Testing of hypothesis, Interval estimation. Properties of estimators, Methods of estimation: Method of moments, Method of maximum likelihood. Some concepts related to Testing of Hypothesis, NeymanPearson lemma, Likelihood-ratio tests.. Some common tests of hypothesis for normal populations, Confidence intervals, Chi-square test for goodness of fit.

MTE-12:

LINEAR PROGRAMMING

4 Credits

Matrices(Addition and Multiplication,transpose, inverse, rank), Vector spaces(Vectors, Linear independence, basis, dimension), Inequalities (Equations, Inequations and their graphs), convex sets(Definition, examples, definition of extreme point, half spaces, hyperplanes, convex combination of finite number of points is a convex set(no proof), definition of convex hull, convex hull of a finite set is the convex combination of its points(no proof). Mathematical formulation of maximization and minimization problems, graphical solution, bounded sets, unbounded sets, alternative optimum. Maximization and minimization problems, graphical solution of problems in three variables, mathematical formulation of problems in more than three variables(maximization problems, minimization problems, General Linear Programming Problem(GLPP) (Slack variables, surplus variables, unrestricted variables, reduction of a GLPP to its standard form), Matrix formulation of the standard form, canonical form of a LPP, reducing a GLPP to its canonical form, types of solutions(feasible solutions, basic solutions, basic feasible solutions), finding basic solutions of a system of simultaneous linear equations. The statement of the following theorems(without proofs) 1. If there is a feasible solution to a LPP, there is also abasic feasible solution. 2. Every basic feasible solution of an LPP corresponds to one of the extreme points the set of feasible solutions. 3. If a minimum or a maximum exists for a LPP, it is attained at one of the extreme points of feasible region. Meaning of algorithm, explanation of the steps in simplex algorithm, examples. Artificial variable method. Definition of the dual of a LPP. Writing the dual of a given GLPP, Dual of the Dual is the primal, Weak duality and strong duality theorems(statements only), illustration of duality theorems through examples, significance of duality. The Transportation Problem, mathematical formulation of Transportation Problems, balanced Transportation Problems, tabular representation of Transportation Problems, special structure of the Transportation Problem. North-West corner Method, MatrixMinima method, checking whether a feasible solution to a Transportation Problem is a basic solution using closed chain rule, degenerate basic feasible solutions, solving balanced Transportation Problem by U-V method, an unbalanced transportation problem, formulation of an Assignment Problem, solving an assignment problem by Hungarian method. Definition of a Game, the pay-off-matrix, Games of pure strategy, Maxmin and Minmax Principle, saddle point of a pay-off matrix, value of the game, solving games of pure strategy, mixed strategies, expected value, algebraic method for 25 2 matrix games of mixed strategies, short cut method for 25 2 matrix games of mixed strategies, graphical solution of m5 2 and 25 m games by graphical method, dominance Property, modified dominance property, using algebraic method when the pay-off matrix is a square matrix, reducing the rectangular matrix game to that of a linear programming problem, fundamental Theorem of rectangular games, important properties of optimal mixed strategies.

MTE-13

DISCRETE MATHEMATICS

4 Credits

Propositions, Logical connectives: Disjunction, Conjunction, Negation, Conditional Connectives, Precedence rule; Logical Equivalence, Logical quantifiers.. What is a proof?; Different methods of proof: Direct proof, Indirect proofs, Counter examples; Principle of induction. Boolean algebras; Boolean expressions; Logic circuits; Boolean functions. Multiplication and Addition Principles; Permutations: Notations, Circular Permutations, Permutation of objects not necessarily distinct; Combinations: Formula for C(n,r), Combination with repetition, The Binomial expansion: Pascal’s formula for C(n,r), Some identities involving binomial coefficients; The multinomial expansion: Applications to combinatorial probability, Elements of classical probability theory, Addition theorem in probability. Integer Partitions: Recurrence Relation for Pnk , Ferrar’s graph, A recurrence relation for the number of partitions, generating function for Pn’s; Distributions: Distinguishable objects into distinguishable containers, Generating function Approach, containers with at most one ohject, Distinguishable objects into indistinguishable containers, Stirling numbers of the second kind, Recurrence relation for Snm, A Generalization of the recurrence relation for Stirling numbers of the second kind, Generating function for Stirling number of second kind, Bell Numbers, Indistinguishable objects into distinguishable containers, Indistinguishable objects into indistinguishable containers. Pigeon-Hole principle; Inclusion and exclusion principle: Application to number theory - Euler’s toitient function, Application to onto maps, Application to Probability, Applications to Derangements. Three recurrent problems; More recurrences; Definitions; Divide and conquer. Generating functions; Exponential generating function; Applications: Combinatorial identities, Linear equations, Partitions, Recurrence relations. Linear homogeneous recurrences; Linear non-homogeneous recurrences; Some other methods; Method of inspection, Method of telescoping sums, Method of iteration, Method of substitution. Graphs; Regular graphs; Subgraphs. Connected graphs, Paths, Circuits and Cycles, Components, Connectivity, Bipartite graphs, Trees. Eulerian graphs; Fleury’s algorithm; Hamiltonian graphs; Travelling salesperson problem. Vertex colourings: Definition and examples; Bounds for chromatic numbers; Planar graphs; When is a graph planar?; Map colouring problem; Edge colourings.

MTE-14

MATHEMATICAL MODELLING

4 Credits

Mathematical Modelling – what and why?, Types of modelling, Limitations of a Mathematical Model. Identifying the Essentials of a problem, Mathematical Formulation (Motion of a simple pendulum, growth of phytoplanktons, motion of a raindrop). Solution of formulated problems (Motion of A Simple Pendulum, phytoplankton growth), Interpretation of the solutions. Free fall of a body, Upward Motion Under Gravity, Simple Harmonic Motion, Projectile Motion. Noewton’s Law of Gravitation, Escape Velocity, Central forces – Basic Concepts, Modelling Planetary Motion, Kepler’s laws, Limitations of the model. Physical Process, Mathematical Model of Plume Rise, Gaussian Model of Dispersion, Applications of Gaussian Model. Modelling Blood Flow problem and Oxygen transfer in Red cells. Formulation, solution, Interpretation and Limitations of the Models. Exponential growth model and Logistic growth model – Formulation, solution, Interpreation and Limitations of the models, Extension of the Logistic Model. Types of Interactions between two species, Prey-Predator and Competing Species Models – Formulation, solution, Interpretation and Limitations of the models. Basic definitions and fundamental concepts, Simple epidemic model, General epidemic model, Recurrent epidemic model without and with Undamped waves – Formulation, Solution, Interpretation and Limitations of the models. Utility, demand, production, cost and supply functions, Market Equilibrium,Monopoly, Duopoly and Oligopoly. Some games and their characteristics, Two-person zero sum game, Co-operative and Non-co-operative game theory – “Prisoner’s Dilemma”, “Battle of Sexes” games, Duopoly, some generalisations, applications and Limitations of the models. Understanding the problem of Investments, Markowitz Model – Return Valuations, risk valuations, diversification, Portfolio Selection, - feasible set, efficient and optimal portfolio, Limitations of the models developed. Queuing – what and why? Basic concepts, structure and techniques, Modelling random phenomena, two queueing models, why forecasting? - Basics and time series analysis, Forecasting Models.