Draft syllabus for b. A/B. Sc. (Honours) in mathematics under Choice Based Credit System (cbcs) Effective from the academic session 2017-2018 sidho-kanho-birsha university purulia-723104 West Bengal
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- Bu səhifədəki naviqasiya:
- CC10 – Ring Theory and Linear Algebra I [Credit: 1+5]
- CC11 – Metric Spaces and Complex Analysis [Credit: 1+5]
- CC12 – Group Theory II, Ring theory-II, Linear Algebra-II [Credit: 1+5]
- CC13 – Numerical Methods Computer Programming [Credit: 1+5]
- Unit 2 [Credit: 2] Introduction
- CC14 – Computer Aided Numerical Statistical Practical (P)
- Department Specific Electives Subjects Syllabus DS1 – Linear Programming
- DS2- Probability and Statistics
- DS5- Differential Geometry
- Generic Elective Courses GE1 – Calculus, Geometry Differential Equation [Credit: 1+5] Unit -1
- Graphical Demonstration (Teaching Aid)
- GE2 – Algebra [Credit: 1+5]
Reference Books
CC10 – Ring Theory and Linear Algebra I [Credit: 1+5] Unit 1
Definition and examples of rings, properties of rings, subrings, integral domains and fields, characteristic of a ring. Ideal, ideal generated by a subset of a ring, factor rings, operations on ideals, prime and maximal ideals. Ring homomorphisms, properties of ring homomorphisms. Isomorphism theorems I, II and III, field of quotients. Unit 2 Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces. Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations. Isomorphisms. Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix. Reference Books
CC11 – Metric Spaces and Complex Analysis [Credit: 1+5] Unit 1 [Credit: 2] Metric spaces: Sequences in metric spaces, Cauchy sequences. Complete Metric Spaces, Cantor’s theorem. Continuous mappings, sequential criterion and other characterizations of continuity. Uniform continuity. Connectedness, connected subsets of R. Compactness: Sequential compactness, Heine-Borel property, Totally bounded spaces, finite intersection property, and continuous functions on compact sets. Homeomorphism. Contraction mappings. Banach Fixed point Theorem and its application to ordinary differential equation. Unit 2 [Credit: 3] Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, and definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy- Goursat theorem, Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and its examples, absolute and uniform convergence of power series. Reference Books
CC12 – Group Theory II, Ring theory-II, Linear Algebra-II [Credit: 1+5] Unit 1
Group actions, stabilizers and kernels, permutation representation associated with a given group action. Generalized Cayley’s theorem. Groups acting on themselves by conjugation, class equation and consequences, conjugacy in Sn, p-groups, Sylow’s theorems and consequences, Cauchy’s theorem, Unit 2 Euclidean Domain, PID, UFD, Polynomial Ring. Unit-3 Inner product spaces and norms, Gram-Schmidt orthogonalisation process, orthogonal complements, Bessel’s inequality, the adjoint of a linear operator. Least Squares Approximation, minimal solutions to systems of linear equations. Normal and self-adjoint operators. Orthogonal projections and Spectral theorem. Reference Books
CC13 – Numerical Methods & Computer Programming [Credit: 1+5] Unit 1 [Credit: 3] Algorithms. Convergence. Errors: Relative, Absolute. Round off. Truncation. Transcendental and Polynomial equations: Bisection method, Newton’s method, Secant method, Regula-falsi method, fixed point iteration, Newton-Raphson method. Rate of convergence of these methods. System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis. LU Decomposition Interpolation: Lagrange and Newton’s methods. Error bounds. Finite difference operators. Gregory forward and backward difference interpolation. Numerical differentiation: Methods based on interpolations, methods based on finite differences. Numerical Integration: Newton Cotes formula, Trapezoidal rule, Simpson’s 1/3rd rule, Simpsons 3/8th rule, Weddle’s rule, Boole’s Rule. Midpoint rule, Composite Trapezoidal rule, Composite Simpson’s 1/3rd rule, Gauss quadrature formula. The algebraic eigenvalue problem: Power method. Approximation: Least square polynomial approximation. Ordinary Differential Equations: The method of successive approximations, Euler’s method, the modified Euler method, Runge-Kutta methods of orders two and four.
CC14 – Computer Aided Numerical & Statistical Practical (P) List of practical C language 1. Calculate the sum 1/1 + 1/2 + 1/3 + 1/4 + ----------+ 1/ N. 2. Enter 100 integers into an array and sort them in an ascending order. 3. Solution of transcendental and algebraic equations by a. Newton Raphson method. b. Fixed point method. 4. Solution of system of linear equations a. Gaussian elimination method b. Gauss-Seidel method 5. Interpolation a. Lagrange Interpolation b. Newton Interpolation 6. Numerical Integration a. Trapezoidal Rule b. Simpson’s one third rule 7. Solution of ordinary differential equations a. Euler method b. Modified Euler method c. Runge Kutta method Department Specific Electives Subjects Syllabus DS1 – Linear Programming Introduction to linear programming problem. Theory of simplex method, graphical solution, convex sets, optimality and unboundedness, the simplex algorithm, simplex method in tableau format, introduction to artificial variables, two‐phase method. Big‐M method and their comparison. Duality, formulation of the dual problem, primal‐dual relationships, economic interpretation of the dual. Transportation problem and its mathematical formulation, northwest‐corner method, least cost method and Vogel approximation method for determination of starting basic solution, algorithm for solving transportation problem, assignment problem and its mathematical formulation, Hungarian method for solving assignment problem. Game theory: formulation of two person zero sum games, solving two person zero sum games, games with mixed strategies, graphical solution procedure, linear programming solution of games.
DS2- Probability and Statistics Sample space, probability axioms, real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function, characteristic function, discrete distributions: uniform, binomial, Poisson, geometric, negative binomial, continuous distributions: uniform, normal, exponential. Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables, conditional expectations, independent random variables, bivariate normal distribution, correlation coefficient, joint moment generating function (jmgf) and calculation of covariance (from jmgf), linear regression for two variables. Chebyshev’s inequality, statement and interpretation of (weak) law of large numbers and strong law of large numbers. Central Limit theorem for independent and identically distributed random variables with finite variance, Markov Chains, Chapman-Kolmogorov equations, classification of states. Random Samples, Sampling Distributions, Estimation of parameters, Testing of hypothesis.
DS3- Number Theory Linear Diophantine equation, prime counting function, statement of prime number theorem, Goldbach conjecture, linear congruences, complete set of residues, Chinese Remainder theorem, Fermat’s Little theorem, Wilson’s theorem. Number theoretic functions, sum and number of divisors, totally multiplicative functions, definition and properties of the Dirichlet product, the Mobius Inversion formula, the greatest integer function, Euler’s phi‐function, Euler’s theorem, reduced set of residues. some properties of Euler’s phi-function. Order of an integer modulo n, primitive roots for primes, composite numbers having primitive roots, Euler’s criterion, the Legendre symbol and its properties, quadratic reciprocity, quadratic congruences with composite moduli. Public key encryption, RSA encryption and decryption, the equation x2 + y2= z2, Fermat’s Last theorem. Reference Books
DS4- Mechanics Co-planar forces. Astatic equilibrium. Friction. Equilibrium of a particle on a rough curve. Virtual work. Forces in three dimensions. General conditions of equilibrium. Centre of gravity for different bodies. Stable and unstable equilibrium. Equations of motion referred to a set of rotating axes. Motion of a projectile in a resisting medium. Stability of nearly circular orbits. Motion under the inverse square law. Slightly disturbed orbits. Motion of artificial satellites. Motion of a particle in three dimensions. Motion on a smooth sphere, cone, and on any surface of revolution. Degrees of freedom. Moments and products of inertia. Momental Ellipsoid. Principal axes. D’Alembert’s Principle. Motion about a fixed axis. Compound pendulum. Motion of a rigid body in two dimensions under finite and impulsive forces. Conservation of momentum and energy. Reference Books
DS5- Differential Geometry Theory of Space Curves: Space curves. Planer curves, Curvature, torsion and Serret-Frenet formula. Osculating circles, Osculating circles and spheres. Existence of space curves. Evolutes and involutes of curves. Theory of Surfaces: Parametric curves on surfaces. Direction coefficients. First and second Fundamental forms. Principal and Gaussian curvatures. Lines of curvature, Euler’s theorem. Rodrigue’s formula. Conjugate and Asymptotic lines. Developables: Developable associated with space curves and curves on surfaces, Minimal surfaces. Geodesics: Canonical geodesic equations. Nature of geodesics on a surface of revolution. Clairaut’s theorem. Normal property of geodesics. Torsion of a geodesic. Geodesic curvature. Gauss-Bonnet theorem. Reference Books
Generic Elective Courses GE1 – Calculus, Geometry & Differential Equation [Credit: 1+5] Unit -1 Hyperbolic functions, higher order derivatives, Leibnitz rule of successive differentiation and its applications, concavity and inflection points, envelopes, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L’Hospital’s rule, applications in business, economics and life sciences. Unit-2 Reduction formulae, derivations and illustrations of reduction formulae, parametric equations, parametrizing a curve, arc length, arc length of parametric curves, area of surface of revolution. Techniques of sketching conics.
Reflection properties of conics, translation and rotation of axes and second degree equations, classification of conics using the discriminant, polar equations of conics. Spheres. Cylindrical surfaces. Central conicoids, paraboloids, plane sections of conicoids, Generating lines, classification of quadrics, Illustrations of graphing standard quadric surfaces like cone, ellipsoid.
Differential equations and mathematical models. General, particular, explicit, implicit and singular solutions of a differential equation. Exact differential equations and integrating factors, separable equations and equations reducible to this form, linear equation and Bernoulli equations, special integrating factors and transformations. Graphical Demonstration (Teaching Aid) 1. Plotting of graphs of function eax + b, log(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b), |ax + b| and to illustrate the effect of a and b on the graph. 2. Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them. 3. Sketching parametric curves (Eg. Trochoid, cycloid, epicycloids, hypocycloid). 4. Obtaining surface of revolution of curves. 5. Tracing of conics in Cartesian coordinates/ polar coordinates. 6. Sketching ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic, paraboloid, and hyperbolic paraboloid using Cartesian coordinates. Reference Books
GE2 – Algebra [Credit: 1+5] Unit -1 [Credit: 3] Polar representation of complex numbers, n-th roots of unity, De Moivre’s theorem for rational indices and its applications. Theory of equations: Relation between roots and coefficients, Transformation of equation, Descartes rule of signs, Cubic and biquadratic equations, special roots, reciprocal equation, binomial equation. Inequality: The inequality involving AM≥GM≥HM, Cauchy-Schwartz inequality. Equivalence relations and partitions, Functions, Composition of functions, Invertible functions, One to one correspondence and cardinality of a set. Well-ordering property of positive integers, Division algorithm, Divisibility and Euclidean algorithm. Congruence relation between integers. Principles of Mathematical Induction, statement of Fundamental Theorem of Arithmetic. Unit -2 [Credit: 2] Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation Ax=b, solution sets of linear systems, applications of linear systems, linear independence. Introduction to linear transformations, matrix of a linear transformation, inverse of a matrix, characterizations of invertible matrices. Subspaces of Rn, dimension of subspaces of Rn, rank of a matrix, Eigen values, Eigen Vectors and Characteristic Equation of a matrix. Cayley-Hamilton theorem and its use in finding the inverse of a matrix.
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