Real Analysis -I
1. Sets and functions: Operations on sets, Functions, Real-valued functions, Equivalence countability, Real numbers, Cantor set, Least upper bounds 2. Sequences of Real Numbers: Definition of sequence and subsequence, Limit of a sequence, Convergent sequences, Monotone sequences, Divergent sequences, Limit superior, Limit inferior, Cauchy sequences. 3. Series of Real numbers: Convergent and divergent series, series with non-negative terms, alternating series, Conditional and Absolute convergence, Rearrangement of series, Tests of absolute convergence, series whose terms form a non-increasing sequence, The class 𝑙2.
1. Riemann Integral: Sets of measure zero, Definition and existence of Riemann integral, properties of Riemann integral, Fundamental theorem of integral calculus, mean value theorems of integral calculus. 2. Improper Integrals: Definition of improper integral of first kind, comparison test, test, absolute and conditional convergence, integral test for convergence of series, definition of improper integral of second kind, Cauchy principal value. 3. Sequences and series of functions: Point wise and uniform convergence of sequences of functions, consequences of uniform convergence, convergence and uniform convergence of series of functions, integration and differentiation of series of functions.
Group Theory 1. Groups: Binary Operations, Isomorphic Binary Structures, Groups. 2. Subgroups: Subgroups, Cyclic Groups. 3. Permutations: Cosets, Direct Product: Groups of Permutations, Orbits, Cycles, Alternating Groups, Cosets and the Theorem of Lagrange, Direct Products. 4. Homomorphisms and Factor Groups: Homomorphisms, Factor Groups, Factor Group Computations and Simple Groups. Ring Theory 1. Rings and Fields: Rings and Fields, Integral Domains, The Fields of Quotients of an Integral Domain, Rings of Polynomials, Factorization of Polynomials over a Field. 2. Ideals and Factor Rings: Homomorphisms and Factor Rings, Prime and Maximal Ideals. Factorization: Unique Factorization Domains , Euclidean Domain Euclidean Domains, Gaussian Integers and Multiplicative Norms
Ordinary Differential Equations
1. Linear Differential Equations with constant coefficients: The auxiliary equations. Distinct roots, repeated roots, Complex roots, particular solution. The operator 1/𝑓 (𝐷) and its evaluation for the functions 𝑥𝑚 , 𝑒𝑎𝑥, 𝑒𝑎𝑥𝑣 & 𝑥𝑣 and the operator 1/(𝐷2 + 𝑎2) acting on 𝑠𝑖𝑛 𝑎𝑥 and 𝑐𝑜𝑠 𝑎𝑥 with proofs. 2. Non-Homogeneous Differential Equations: Method of undetermined coefficients, Method of variation of parameters, Method of reduction of order, The use of a known solution to find another. 3. Power series solutions: Introduction and review of power series, Linear equations and power series, Convergence of power series,Ordinary points and regular singular points. 4. System of First-Order Equations: Introductory remarks, linear systems, homogeneous linear systems with constant Coefficients, Distinct roots, repeated roots, Complex roots.
1. Modeling with Linear Programming: Two variable LP Model, Graphical LP solution, Selected LP Applications, Graphical Sensitivity analysis. 2. The Simplex Method: LP Model in equation form, Transition from graphical to algebraic solutions, the simplex method, Artificial starting solutions. Duality: Definition of the dual problem, Primal dual relationship. 3. Transportation Model: Definition of the Transportation model. The Transportation algorithm. 4. The Assignment Model: The Hungarian method, Simplex explanation of the Hungarian method.
1. Introductory Concepts: Introduction to computer. Computer Characteristics. Types of Programming Languages. Introduction to C. 2. C Fundamentals: The character set. Identifier and keywords. Data types. Constants. Variables and arrays. Declarations. Expressions. Statements. Symbolic constants. 3. Operators and Expressions: Arithmetic operators. Unary operators. Relational and Logical operators. Assignment operators. Conditional Operator. Library functions. 4. Data Input and Outputs: Preliminaries. Single character input-getchar() function. Single character outputputchar() function. Writing output data-printf function. Formatted input-output. Get and put functions. 5. Preparing and Running a Program: Planning and writing a C Program. Compiling and Executing the Program. 6. Control Statements: Preliminaries. The while statement. The do-while statement. The for statement. Nested loops. The if-else statement. The switch statements. The break statement. The continue statement. The comma operator. 7. Functions: A brief overview. Defining a function. Accessing a function. Passing arguments to a function. Specifying argument data types. Function prototypes ,Recursion. 8. Arrays: Defining an array. Processing an array. Passing arrays to a function. Multidimensional arrays. Arrays and strings.
C programming II
1. Program Structures: Storage classes. Automatic variables. External variables, Static variables. 2. Pointers: Fundamentals. Pointer declarations. Passing pointer to a function. Pointer and one dimensional arrays. Dynamic memory allocation. Operations on pointers. Pointers and multidimensional arrays. Array of pointers. Pointer to function. Passing functions to other functions. More about pointer declarations. 3. Structures and Unions: Defining a structure. Processing a structure. Userdefined data types (typedef ). Structures and pointers. Passing structure to a function. Self-referential structures, Unions. 4. Data Files: Opening and closing a data file. Creating a data file. Processing a data file. Unformatted data files. 5. Low-Level Programming: Bitwise operators. Register variables. Enumerations. Macros. Command line arguments. The C processor. Lattice Theory 1. Ordered Sets a. Ordered sets. b. Examples from social science and computer science. c. Diagrams : the art of drawing ordered sets. d. Constructing and de-constructing ordered sets. e. Down-sets and up-sets. f. Maps between ordered sets. 2. Lattices and Complete Lattices a. Lattice as ordered sets. b. Lattices as algebraic structures. c. Sublattices, products and homomorphisms. d. Ideals and Filters. e. Complete lattices and Intersection-structures. f. Chain conditions and completeness. g. Join-irreducible elements. 3. Modular, distributive and Boolean Lattices a. Lattices satisfying additional identities. b. The characterization Theorems of Modular and Distributive lattices. c. Boolean lattices and Boolean algebras. d. Boolean terms and disjunctive normal form.
1. Mathematical models in economics Introduction, a model of the market, market equilibrium and excise tax. 2. The elements of finance and the cobweb model: Interest and capital growth, income generation, the interval of compounding, stability of market equilibrium, the general linear case and economic interpretation. 3. Introduction to optimization: Profit maximization, critical points, optimization in an interval and infinite intervals. 4. The derivative in economics: Elasticity of demand, profit maximization again, competition versus monopoly, the efficient small firm, startup and breakeven points. 5. Linear equations : Making money with matrices, a two-industry ‘economy’, arbitrage portfolios and state prices, IS-LM analysis. 6. The input-output model: An economy with many industries and the technology matrix.
1. Divisibility : Divisibility in integers, Division Algorithm, GCD, LCM, Fundamental theorem of Arithmetic, Infinitude of primes, Mersenne Numbers and Fermat Numbers. 2. Congruences Properties of Congruences, Residue classes, complete and reduced residue system, their properties, Fermat’s theorem. Euler’s theorem, Wilson’s theorem, has a solution if and only if p = 2 or , where p is a prime. Linear Congruences of degree 1, Chinese remainder theorem. 3. Greatest integer function: Arithmetic functions Euler’s function, the number of divisors d(n), sum of divisors Ω . Multiplicative functions, function, inversion formula. 4. Quadratic Reciprocity: Quadratic residues, Legendre’s symbol. Its properties, Law of quadratic reciprocity. 5. Diophantine Equations : Diophantine Equations ax + by = c and Pythagorean triplets.
1. Introductory Concepts Definition and examples of metric spaces, open spheres and closed spheres, neighbourhoods, open sets, equivalent Metrics, interior points, closed sets, limit points and isolated points, closure of a set, boundary points, distance between sets and diameter of a set, subspace of a metric space, product metric spaces. 2. Completeness Convergent sequences, Cauchy sequences, complete spaces, dense sets and nowhere dense sets (only definition) 3. Continuous Functions: Definition and characterizations, extension theorem, uniform continuity, homeomorphism 4. Compactness Compact spaces, sequential compactness, equivalence of compactness and sequential compactness, compactness and finite intersection property, continuous functions and compact spaces. 5. Connectedness Separated sets, disconnected and connected sets.
1. Complex Numbers Sums and products, Basic algebraic properties, Further properties, Vectors and Moduli, Complex Conjugates, Exponential Form, Products and powers in exponential form, Arguments of products and quotients, Roots of complex numbers, Examples, Regions in the complex plane. 2. Analytic functions Functions of Complex Variables, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives, Differentiation formulas, Cauchy- Riemann Equations, Sufficient Conditions for differentiability, Polar coordinates, Analytic functions, Harmonic functions. 3. Elementary Functions The Exponential functions, The Logarithmic function, Branches and derivatives of logarithms, Some identities involving logarithms, Complex exponents, Trigonometric functions, Hyperbolic functions. 4. Integrals Derivatives of functions, Definite integrals of functions, Contours, Contour integral, Examples, Upper bounds for Moduli of contour integrals, Anti-derivatives, Examples, Cauchy-Groursat’s Theorem (without proof), Simply and multiply Collected domains. Cauchy integral formula, Derivatives of analytic functions. Liouville’s Theorem and Fundamental Theorem of Algebra. 5. Series Convergence of sequences and series, Taylor’s series, Laurent series (without proof), examples. 6. Residues and Poles Isolated singular points, Residues, Cauchy residue theorem, residue at infinity, types of isolated singular points, residues at poles, zeros of analytic functions, zeros and poles.
Partial Differential Equations
1. Ordinary Differential Equations in More Than Two Variables (a) Surface and Curves in Three Dimensions (b) Simultaneous Differential Equations of the First Order and the First Degree in Three Variables. (c) Methods of solution of (d) Orthogonal Trajectories of a System of curves on a Surface. (e) Pfaffian Differential Forms and Equations. (f) Solution of Pfaffian Differential Equations in Three Variables 2. First Order Partial Differential Equations: (a) Genesis of First Order Partial Differential Equations. (b) Classification of Integrals. (c) Linear Equations of the First Order. (d) Pfaffian Differential Equations. (e ) Compatible Systems. (f) Charpit’s Method. (g) Jacobi’s Method. (h) Integral Surfaces through a given curve. (i) Quasi-Linear Equations.
1. Network Models CPM and PERT, Network representation, Critical Path Computations, Construction of the time schedule, Linear programming formulation of CPM, PERT calculations. 2. Decision Analysis and Games Decision under uncertainty, Game theory, Some basic terminologies, Optimal solution of two person zero sum game, Solution of mixed strategy games, graphical solution of games, linear programming solution of games. 3. Replacement and Maintenance Models Introduction, Types of failure, Replacement of items whose efficiency deteriorates with time. 4. Sequencing Problems Introduction, Notation, terminology and assumptions, processing n jobs through two machines, processing n jobs through three machines. 5. Classical Optimization Theory Unconstrained problems, Necessary and sufficient conditions, Newton Raphson method, Constrained problems, Equality constraints (Lagrangian Method Only).
1. An Introduction to Graphs The definition of a Graph, Graphs and Models, More Definitions, Vertex Degree, Sub graphs, Paths and Cycles, The Matrix Representation of Graphs, Fusion 2. Trees and Connectivity Definition and Simple Properties, Bridges, Spanning Trees, Connector Problems, Shortest Path Problems, Cut Vertices and Connectivity. 3. Euler Tours and Hamiltonian Cycles Euler Tours, The Chinese Postman Problem, Hamiltonian Graphs, The Travelling Salesman Problem. For school level math training please visit