cotton college state universityccsu.ac.in/admin/gallery/math pg_syllabus_ccsu_july28_r.pdf ·...

1 Version 1.0, 2014

COTTON COLLEGE STATE UNIVERSITY

DEPARTMENT OF MATHEMATICS

Postgraduate Mathematics Syllabus

DISTRIBUTION OF PAPERS/CREDITS (L+T+P format)

Semester – I

Paper Code Paper Name Credits

MTH 701C Analysis -1 3 + 1 + 0

MTH 702C Linear Algebra 3 + 1 + 0

MTH 703C Differential Equations -1 3 + 1 + 0

MTH 704C Numerical Analysis 3 + 1 + 0

An elective paper will be offered by the Department from the list of elective papers given here.

2 + 1 + 0

Semester – II


MTH 801C Analysis – 2 3 + 1 + 0

MTH 802C Abstract Algebra 3 + 1 + 0

MTH 803C Mathematical Methods 3 + 1 + 0

MTH 804C Topology 3 + 1 + 0


2 + 1 + 0

2 Version 1.0, 2014

Semester – III


MTH901C Complex Function Theory 3 + 1 + 0

MTH 902C Functional Analysis 3 + 1 + 0

MTH 903C Differential Equations 2 3 + 1 + 0

MTH904C Tensors and Mechanics 3 + 1 + 0


2 + 1 + 0

Semester - IV


MTH1001C Elementary Number Theory 3 + 1 + 0

MTH1002C Computer Programming 2 + 0 + 2

MTH 1003C Discrete Mathematics 3 + 1 + 0

A special paper to be offered by the Department from the list given here.

3 + 1 + 0


2 + 1 + 0

3 Version 1.0, 2014

SEMESTER- I Paper: MTH 701C

ANALYSIS - 1 Credits: 4 (3+1+0)

UNIT I: Brief review of sets, relations and functions. Finite and infinite sets, countable and uncountable sets, Schröder-Bernstein theorem, Ordered fields, least upper bound property, the field of real numbers, Archimedean property, density of rational numbers, existence of nth root of positive real numbers, exponential and logarithm, the extended real number system, the complex field.

UNIT II: Numerical sequences and their convergence, bounded sequences, Cauchy sequences, construction of real numbers using Cauchy sequences; series of complex numbers, convergence of series, series of nonnegative terms, the number e, the root and ratio tests, limit supremum and limit infimum, power series, summation by parts, absolute convergence, addition and multiplication of series, rearrangements (statement only).

UNIT III: Euclidean spaces, metric spaces, open and closed sets, limit points, interior points, compact spaces; statements only of the following: nested interval theorem, Heine-Borel theorem, and Bolzano-Weierstrass’ theorem.

UNIT IV: Limits of functions, continuous functions, continuity and compactness, uniform continuity, connected sets, connected subsets of real numbers, continuity and connectedness, intermediate value theorem; discontinuities and their classifications, monotonic functions, infinite limits and limits at infinity.

UNIT-V: Differentiation of real-valued functions and its elementary properties; mean value theorem; Taylor’s theorem; differentiation of vector-valued functions; elementary properties of Riemann integral (brief review); integration of vector-valued functions. Textbooks: 1. Naïve Set Theory (3rd edition) – P. R. Halmos, D. Van Nostrand Co., Inc, Princeton, New Jersey, 2002. 2. Principles of Mathematical Analysis (5th edition) – W. Rudin, McGraw Hill Kogakusha Ltd., 2004. 3. An Introduction To The Theory of Groups (4th edition) – J. J. Rotman, Allyn and Bacon, Inc., Boston, 2002. Reference books: 1. Mathematical Analysis (5th edition) – T. Apostol, Addison-Wesley; Publishing Company, 2001.

4 Version 1.0, 2014

2. Introduction to Real Analysis (3rd edition) – R. G. Bartle and D. R. Sherbert, John Wiley & Sons, Inc., New York, 2000. 3. A First Course in Abstract Algebra (4th edition) – J. B. Fraleigh, Narosa Publishing House, New Delhi, 2002. 4. Contemporary Abstract Algebra (4th edition) – J. A. Gallian, Narosa Publishing House, New Delhi, 1999. 5. Basic Real Analysis – H.H. Sohrab, Birkhäuser (2003).


LINEAR ALGEBRA Credits: 4 (3+1+0)

UNIT-I: Vector spaces, linear independence; linear transformations, matrix representation of a linear transformation; isomorphism between the algebra of linear transformations and that of matrices;

UNIT-II: Similarity of matrices and linear transformations; trace of matrices and linear transformations, characteristic roots and characteristic vectors, characteristic polynomials, relation between characteristic polynomial and minimal polynomial; Cayley-Hamilton theorem (statement and illustrations only); diagonalizability, necessary and sufficient condition for diagonalizability;

UNIT-III: Projections and their relation with direct sum decomposition of vector spaces; invariant subspaces; primary decomposition theorem, cyclic subspaces; companion matrices; a proof of Cayley-Hamilton theorem; triangulability; canonical forms of nilpotent transformations; Jordan canonical forms; rational canonical forms.

UNIT-IV: Inner product spaces, properties of inner products and norms, Cauchy-Schwarz inequality; orthogonality and orthogonal complements, orthonormal basis, Gram-Schmidtprocess; adjoint of a linear transformation; Hermitian, unitary and normal transformations and their diagonalizations.

UNIT-V: Forms on inner product spaces and their matrix representations; bilinear forms; Hermitian forms; symmetric bilinear forms; orthogonal diagonalization of real quadratic forms. Textbooks:

5 Version 1.0, 2014

1. Linear Algebra (2nd edition) – K. Hoffman and R. Kunze, Prentice Hall of India Pvt. Ltd., New Delhi, 2000. 2. First Course in Linear Algebra – P. B. Bhattacharya, S. K. Jain and S. R. Nagpal, Wiley Eastern Ltd., New Delhi, 2000. Reference books: 1. Topics in Algebra (4th edition) – I. N. Herstein, Wiley Eastern Limited, New Delhi, 2003. 2. Linear Algebra – G. E. Shilov, Prentice Hall, 1998. 3. Finite Dimensional Vector Spaces – P. R. Halmos, Van Nostrand Inc., 1965. 4. Introduction to Matrices and Linear Transformations (3rd edition) – D. T. Finkbeiner, D.B. Taraporevala, Bombay, 1990. 5. Linear Algebra, A Geometric Approach – S. Kumaresan, Prentice-Hall of India Pvt. Ltd., New Delhi, 2001.


DIFFERENTIAL EQUATIONS - 1 Credits: 4 (3+1+0)

UNIT I: Linear equations with constant coefficients; the second and higher order homogeneous equation; initial value problems for second order equations; existence theorem; uniqueness theorem; linear dependence and independence of solutions; the Wronskian and linear independence; a formula for the Wronskian; the non-homogeneous equation of order two.

UNIT II: Linear equations with variable coefficients, initial value problems for the homogeneous equations; existence theorem; uniqueness theorem; solutions of homogeneous equations; the theorem on n linearly independent solutions; the Wronskian and linear independence;

UNIT III: Existence and uniqueness of solutions – introduction; equations with variable separated; exact equations, Lipschitz condition; non-local existence of solutions; uniqueness of solutions; existence and uniqueness theorem for first order equations; statement of existence and uniqueness theorem for the solutions of ordinary differential equation of order n.

UNIT IV: Initial value problems for the homogeneous equations; solutions of homogeneous equations; Wronskian and linear independence; non-homogeneous equations; homogeneous equations with analytic coefficients; Legendre equation, justification of power series method; Legendre polynomials and Rodrigues’ formulae.

UNIT V:

6 Version 1.0, 2014

Linear equations with regular singular points – introduction; Euler equation; second order equations with regular singular points – example and the general case, convergence proof, exceptional cases; Bessel equation; regular singular points at infinity.

Textbooks: 1. An Introduction to Ordinary Differential Equations – E. A. Coddington, Prentice-Hall of India Private Ltd., New Delhi, 2001 . 2. Spherical Harmonics – T. M. Mac Robert, Pergamon Press, 1967. Reference books: 1. Elementary Differential Equations (3rd Edition) – W. T. Martain and E. Relssner, Addison Wesley Publishing Company, inc., 1995. 2. Theory of Ordinary Differential Equations – E. A. Codington and N. Levinson, Tata McGraw hill Publishing co. Ltd. New Delhi, 1999. 3. Differential Equations, Dynamical Systems and an Introduction to Chaos – M.W. Hirsch, S. Smale, and R.L. Devaney, Elsevier (2004).


NUMERICAL ANALYSIS Credits: 4 (3+1+0)

UNIT I A brief introduction to algebraic and transcendental equations and their roots; direct and iterative methods for determination of roots of these equations; initial approximations; bisection method, secant method, Regula-Falsi method, Newton-Raphson method for determination of roots of algebraic and transcendental equations; error analysis, rate of convergence and algorithm for each of these methods.

UNIT II A brief introduction to systems of linear algebraic equations and their solutions, eigenvalue problem and its solution; direct and iterative methods; forward and backward substitution method; Cramer’s rule; Gauss elimination method; Gauss-Jordan elimination method; Gauss- Jacobi iteration method; Gauss-Seidel iteration method; power method for eigenvalue problem; iterative method for matrix inversion; error analysis, rate of convergence and algorithm for each of these methods.

UNIT III Lagrange and Newton interpolation; Lagrange interpolating polynomial and Newton divided differences interpolating polynomial; linear interpolation; Newton’sdivided difference interpolation and its generalizations; finite difference operators; relation between differences and derivatives; Gregory-Newton forward and backward difference interpolation; truncation error bounds and algorithm.

7 Version 1.0, 2014

UNIT IV Differentiation and integration; numerical differentiation; methods based on linear and quadratic interpolation with error of approximation; methods based on finite differences; optimum choice of step length; numerical integration; methods based on interpolation; determination of the error term; trapezoidal rule; Simpson’s rule; error of integration; algorithms for numerical differentiation and integration.

UNIT V Ordinary differential equations and their numerical solutions; initial value problems; error estimates; Euler-Richardson method, Runge-Kutta methods and PredictorCorrector method; error analysis and algorithm for each of these methods; partial differential equations; finite- difference method with error analysis and algorithm. Textbooks: 1. Introduction to Numerical Analysis (2nd edition) – K. E. Atkinson, John Wiley, 1989. 2. Numerical Methods for scientific and Engineering computation – M. K. Jain, S. R. K. Iyenger and R. K. Jain, New Age international publishers, New Delhi, 2003. 3. Fundamental of Computer Numerical Analysis – M. Friedman and A. Kandel,CRC Press, Boca Raton, 1993. 4. Applied Numerical Analysis (5th edition) – C. F. Gerald and P. O. Wheatley, Addison-Wesley, New York, 1998. Reference books: 1. Elementary Numerical Analysis: An Algorithmic Approach (3rd edition) – S. D. Conte and C. de Boor, McGraw Hill, New York, 1980. 2. Numerical Mathematical Analysis – J. B. Scarborough, Oxford & IBH Publishing Co., 2001. 3. Computer Oriented Numerical Analysis – V. Rajaraman, Prentice-Hall of India Pvt.

SEMESTER- II Paper: MTH 801C

ANALYSIS - 2 Credits: 4 (3+1+0)

UNIT-I: Sequences of functions, pointwise and uniform convergence; uniform convergence and continuity; uniform convergence and integration; uniform convergence and differentiation; nowhere differentiable functions; Statement of Stone-Weierstrass’ theorem for a real and complex-valued functions on an interval.

UNIT-II: Directional derivatives; derivatives of functions of several variables and their interrelationship; chain rule; mean value theorem; higher order partial derivatives; equality of mixed partial derivatives, Schwarz lemma; Taylor’s theorem.

8 Version 1.0, 2014

UNIT-III: Injective mapping theorem, surjective mapping theorem, inverse function theorem and implicit function theorem of functions of two and three (for analogy) variables; extremum problems with and without constraints of functions of two and three (for analogy) variables.

UNIT-IV: σ rings of sets, additive, countably additive, regular set functions, outer measures on power set of reals, measurable spaces, Lebesgue measure, measurable functions and their properties.

UNIT-V: Lebesgue integral, Lebesgue integrable functions, properties of integrals, Lebesgue’s monotone convergence theorem, Fatou’s lemma, Lebesgue’s dominated convergence theorem, integration of complex valued functions, functions of class L2, Fourier series, Riesz-Fischer theorem. Textbooks: 1. Principles of Mathematical Analysis (5th edition) – W. Rudin, McGraw Hill Kogakusha Ltd., 2004. 2. Mathematical Analysis (5th edition) – T. Apostol, Addison-Wesley; Publishing Company, 2001. 3. The Elements of Real Analysis (3rd edition) – R. G. Bartle, Wiley International Edition, 1994. Reference books: 1. Advanced Calculus (4th Edition) – R.C. Buck & E.F. Buck, McGraw Hill Book Company, 1999. 2. Introduction to Topology and Modern Analysis (4th edition) – G. F. Simmons, McGraw Hill Kogakusha Ltd., 2000. 3. Introduction to Real Analysis (3rd edition) – R. G. Bartle and D. R. Sherbert, John Wiley & Sons, Inc., New York, 2000.


ABSTRACT ALGEBRA Credits: 4 (3+1+0)

UNIT I: A brief review of groups, their elementary properties and examples, subgroups, cyclic groups, homomorphism of groups and Lagrange’s theorem; permutation groups, permutations as products of cycles, even and odd permutations, normal subgroups, quotient groups; isomorphism theorems, correspondence theorem.

UNIT II: Group action; Cayley's theorem, group of symmetries, dihedral groups and their elementary properties; orbit decomposition; counting formula; class equation, consequences for p-groups; Sylow’s theorems (proofs using group actions).

9 Version 1.0, 2014

UNIT III: Applications of Sylow’s theorems, conjugacy classes in Sn and An, simplicity of An. Direct product; structure theorem for finite abelian groups; invariants of a finite abelian group (Statements only).

UNIT IV: Basic properties and examples of ring, domain, division ring and field; direct products of rings; characteristic of a domain; field of fractions of an integral domain; ring homomorphisms (always unitary); ideals; factor rings; prime and maximal ideals, principal ideal domain; Euclidean domain; unique factorization domain.

UNIT V: A brief review of polynomial rings over a field; reducible and irreducible polynomials, Gauss’ theorem for reducibility of f(x) ∈ Z[x]; Eisenstein’s criterion for irreducibility of f(x) ∈ Z[x] over Q, roots of polynomials; finite fields of orders 4, 8, 9 and 27 using irreducible polynomials over Z2 and Z3. Textbooks: 1. Basic Abstract Algebra (3rd edition) – P.B. Bhattacharya, S. K. Jain and S. R. Nagpal, Cambridge University Press, 2000. 2. Basic Algebra I (3rd edition) – N. Jacobson, Hindustan Publishing corporation, New Delhi, 2002. 3. Contemporary Abstract Algebra (4th edition) – J. A. Gallian, Narosa Publishing House, New Delhi, 1999. Reference books: 1. Topics in Algebra (4th edition) – I. N. Herstein, Wiley Eastern Limited, New Delhi, 2003. 2. A First Course in Abstract Algebra (4th edition) – J. B. Fraleigh, Narosa Publishing House, New Delhi, 2002. 3. Abstract Algebra – D.S. Dummit, R.M. Foote, John Wiley&Sons (2003)


MATHEMATICAL METHODS Credits: 4 (3+1+0)

Unit I: Fredholm Integral Equations Definition of Integral Equation, Eigen values and Eigen functions: Reduction to a system of algebraic equations, Reduction of ordinary differential equations into integral equations. Fredholm integral equations with separable kernals, Method of successive approximations, Iterative scheme for Fredholm Integral equations of second kind, Conditions of Uniform convergence and uniqueness of series solution.

10 Version 1.0, 2014

Unit II: Voltera Integral Equations Voltera Integral Equations of second kind, Resolvant kernal of Voltera equation and its results, Application of iterative scheme to Voltera integral equation of the second kind. Convolution type kernals.

Unit III: Fourier Transform Fourier Integral Transform. Properties of Fourier Transform, Fourier sine and cosine transform, Application of Fourier transform to ordinary and partial differential equations of initial and boundary value problems. Evaluation of definite integrals.

Unit IV: Laplace Transform Basic properties of Laplace Transform, Convolution theorem and properties of convolution, Inverse Laplace Transform, Application of Laplace Transform to solution of ordinary and partial differential equations of initial boundary value problems. Evaluation of definite integrals.

Unit V: Calculus of Variation with one independent variable Basic ideas of calculus of variation, Euler's equation with fixed boundary of the functional

I[푦(푥)] = 푓 (푥,푦, 푦 )푑푥

containing only the first order derivative of the only dependent variable with respect to one independent variable. Variational problems with functionals having higher order derivatives ot the only dependent variable, applications.

Unit VI: Calculus of Variation with several independent variables Variational problems with functionals dependent on functions of several independent variables having first order derivatives. Variational problems in parametric form, variational problems with subsidiary condition (simple case only), Isoperimetric problems, Applications.

Textbooks: 1. R.P. Kanwal: Linear Integral Equations, Theory and Techniques, Academic Press, New York, 1971. 2. M.R. Spiegel: Theory and Problems of Laplace Transform. 3. A.S. Gupta: Calculus of Variation with Applications, Prentice-Hall of India (1999). Reference Books: 1. S.G. Mikhlin: Linear Integral Equations, (Trans.) Hindustan Book Agency, 1960. 2. Hilderbrand: Methods of Applied Mathematics. 3. Raisinghania: Integral Transforms. 4. R.Courant and D. Hilbert: Methods of Mathematical Physics, Vol-I; Wiley Interscience, New York, 1953.

11 Version 1.0, 2014


TOPOLOGY Credits: 4 (3+1+0)

UNIT I: Definition and examples of topological spaces; basis and sub basis; order relations, dictionary order, order topology; subspace topology; Kuratowski’s closure axioms.

UNIT II: Continuity and related concepts; product topology; quotient topology; a brief introduction to minimal uncountable well ordered set SΏ; countability axioms; Lindelof spaces and separable spaces.

UNIT III: Connected spaces, generation of connected sets; component, path component; local connectedness, local path-connectedness.

UNIT IV: Compact spaces; limit point compact and sequentially compact spaces; locally compact spaces; one point compactification; finite product of compact spaces, statement of Tychonoff's theorem.

UNIT V: Separation axioms; Urysohn’s lemma; Tietze’s extension theorem; statement of Urysohn’s metrization theorem. Textbooks: 1. Topology, a first course – J. R. Munkres, Prentice-Hall of India Ltd., New Delhi, 2000. 2. General Topology – J. L. Kelley, Springer Verlag, New York, 1990. 3. An introduction to general topology (2nd edition) – K. D. Joshi, Wiley Eastern Ltd., New Delhi, 2002. Reference books: 1. General Topology – J. Dugundji, Universal Book Stall, New Delhi, 1990. 2. Foundations of General Topology – W. J. Pervin, Academic Press, New York, 1964. 3. General Topology – S. Willard, Addison-Wesley Publishing Company, Massachusetts, 1970. 4. Basic Topology – M.A. Armstrong, Springer International Ed. 2005.

12 Version 1.0, 2014

SEMESTER- III Paper: MTH 901C

COMPLEX FUNCTIONAL THEORY Credits: 4 (3+1+0)

UNIT I: Brief survey of formal power series, radius of convergence of power series, exponential, cosine and sine, logarithm functions introduced as power series, their elementary properties.

UNIT II: Integration of complex-valued functions and differential 1-forms along a piecewise differentiable path, primitive , local primitive and primitive along a path of a differential 1form, homotopic pats, simply connected domains, index of a closed path, holomorphic functions, Cauchy’s theorem and its corollaries.

UNIT III: Cauchy’s integral formula, Taylor’s expansion of holomorphic functions, Cauchy's estimate; Liouville's theorem; fundamental theorem of algebra; zeros of an analytic function and related results; maximum modulus theorem; Schwarz’ lemma.

UNIT IV: Laurents’s expansion of a holomorphic function in an annulus, singularitiesof a function, removable singularities, poles and essential singularities; extended plane and stereographic projection, residues, calculus of residues; evaluation of definite integrals; argument principle; Rouche's Theorem.

UNIT V: Complex form of equations of straight lines, half planes, circles, etc., analytic (holomorphic) function as mappings; conformal maps; Mobius transformation; cross ratio; symmetry and orientation principle; examples of images of regions under elementary analytic function.

Textbooks: 1. Functions of one complex variable – J. B. Conway, Springer International Student edition, Narosa Publishing House, New Delhi, 2000. 2. Elementary Theory of Analytic Functions of one or several complex variables – H. Cartan, Courier Dover Publications, New York, 1995. Reference books: 1. Complex Analysis (2nd Edition) – L. V. Ahlfors, McGraw-Hill International Student Edition, 1990. 2. Complex Variables and applications – R. V. Churchill, McGraw-Hill, 1996. 3. An Introduction to the Theory of functions of a complex Variable – E. T. Copson, Oxford university press, 1995. 4. An Introduction to Complex Analysis – A. R. Shastri, Macmillan India Ltd., 2003. 5. Complex Variables and Applications – S. Ponnusamy, and H. Silverman, Birhkäuser, 2006.

13 Version 1.0, 2014


FUNCTIONAL ANALYSIS Credits: 4 (3+1+0)

UNIT I:

Classical Banach spaces, Lp spaces; Holder’s inequality, Minkowski’s inequality; convergence and completeness; Riesz-Fischer theorem, bounded linear functional on Lp spaces, Riesz representation theorem.

UNIT II: General Banach spaces – definition and examples; continuous linear transformations between normed linear spaces; Hahn-Banach theorem and its consequences.

UNIT III: Embedding of a normed linear space in its second conjugate space; strong and weak topologies; open mapping theorem; closed graph theorem; uniform boundedness theorem; conjugate of an operator.

UNIT IV: Hilbert’s space, examples and simple properties, orthogonal complements, orthonormal set, Bessel’s inequalities, complete orthonormal sets, Gram-Schmidt orthogonalization process, self adjoint operators.

UNIT V: Normal and unitary operators, projections, spectrum of an operator, spectral theorem for a normal operator on a finite dimensional Hilbert space.

Textbooks: 1. Real Analysis (4th edition) – H. L. Royden, Macmillan Publishing co. inc, New York, 1999. 2. Introduction to Topology and Modern Analysis (4th edition) – G. F. Simmons, Tata McGraw -Hill Ltd., 2004. Reference books: 1. Functional Analysis – W. Rudin, Tata McGraw hill Book Company, 1974 2. Functional Analysis – B. V. Limaye, Willy Eastern Ltd., 1991. 3. First course in Functional Analysis – C.Goffman and G. Pedrick, Prentice-Hall of India Pvt. Ltd, New Delhi, 1974.

14 Version 1.0, 2014


DIFFERENTIAL EQUATIONS - 2 Credits: 4 (3+1+0)

UNIT I: Definition of PDE, origin of first-order PDE; determination of integral surfaces of linear first order partial differential equations passing through a given curve; surfaces orthogonal to given system of surfaces; non-linear PDE of first order, Cauchy’s method of characteristic; compatible system of first order PDE; Charpit’s method of solution, solutions satisfying given conditions, Jacobi’s method of solution.

UNIT II: Origin of second order PDE, linear second order PDE with constant coefficients, linear second order PDE with variable coefficients; characteristic curves of the second order PDE; Monge’s method of solution of non-linear PDE of second order.

UNIT III: Separation of variables in a PDE; Laplace’s equation, elementary solutions of Laplace’s equations; families of equipotential surfaces.

UNIT IV: Wave equation, the occurrence of wave equations, elementary solutions of one-dimensional wave equation; vibrating membranes, three dimensional problems.

UNIT V: Diffusion equation, resolution of boundary value problems for diffusion equation, elementary solutions of diffusion equation, separation of variables

Text Book: 1. Elements of Partial Differential Equation (3rd edition) – I. N. Sneddon, McGraw Hill Book Company, 1998. Reference Book: 1. Partial Differential Equations (2nd edition) – E. T. Copson, Cambridge University Press, 1995.


MECHANICS AND TENSORS Credits: 4 (3+1+0)

UNIT I: Motion in three dimensions, velocity and acceleration in cylindrical and spherical polar coordinates, motion on cylindrical, spherical and conical surfaces.

15 Version 1.0, 2014

UNIT II: Motion of a rigid body under impulsive forces; Application of Principle of virtual work in impulsive motions; Carnot’s theorem; Kelvin’s theorem and Bertrand’s theorem.

UNIT III: Motion of a rigid body about a fixed point; Euler’s Geometrical and Dynamical systems; Motion under no external pressure.

UNIT IV: General coordinates, Lagrange’s equation of motion for finite and impulsive forces in holonomic systems. Case of conservative forces and theory of small oscillation.

UNIT V: Hamilton’s equation of motion, Variational methods. Hamilton’s principle, and principle of least action.

Tensors

UNIT VI: Transformation laws of covariant tensor, Mixed tensors, Rank of tensors, symmetric and anti-symmetric tensors and related theorems, Algebraic operations on tensors, contraction, Inner and outer product of tensors, Quotient law, group property of tensors, Christoffel’s brackets of

first and second kinds, their properties, Riemannian metric, Definitions of metric tensors 푔 and 푔 Transformation laws of Christoffel brackets. Proof of

, + , = 휕푔휕푥

푎푛푑 = 휕 log ±푔

휕푥

UNIT VII:

Covariant derivatives of tensors 퐴 , 퐴 , 퐴 ,퐴 ,퐴 ,Generalizations, Covariant derivatives of metric tensors and scalar invariant function, Application in problems. Angle between two vectors, Laplacian in tensor form.

Text Book: 1. Dynamics of a Particle in rigid bodies, S. L. Loney; CUP 2. An Introduction to Riemannian Geometry and Tensor Calculus by C.E. Weatherburn; CUP, 1950. Reference Book: 1. Dynamics, Part –II, Ramsey.

16 Version 1.0, 2014

2. Textbooks of Dynamics by F. CHorlton, Von Nostrand. 3. Classical Dynamics by S.N. Gupta. 4. Riemannian Geometry by L.P. Eisenhart, Princeton University Press (1949). 5. Tensor Calculus and Riemannian Geometry by D.C. Agarwal.

SEMESTER- IV Paper: MTH 1001C

ELEMENTARY NUMBER THEORY Credits: 4 (3+1+0)

UNIT I:

Divisibility; Euclidean algorithm; primes; congruences; Fermat’s theorem, Euler’s theorem and Wilson’s theorem; Fermat’s quotients and their elementary consequences; solutions of congruences; Chinese remainder theorem; Euler’s phi-function.

UNIT II: Congruence modulo powers of prime; power residues; primitive roots and their existence; quadratic residues; Legendre symbol, Gauss’ lemma about Legendre symbol; quadratic reciprocity law; proofs of various formulations; Jacobi symbol.

UNIT III: Greatest integer function; arithmetic functions, multiplicative arithmetic functions (elementary ones); Mobius inversion formula; convolution of arithmetic functions, group properties of arithmetic functions; recurrence functions; Fibonacci numbers and their elementary properties.

UNIT IV: Diophantine equations – solutions of ax + by = c, x^2 + y^2 = z^2 , x^4 + y^4 = z^2 ; properties of Pythagorean triples; sums of two , four and five squares; assorted examples of diophantine equations.

UNIT V: Simple continued fractions, finite and infinite continued fractions, uniqueness, representation of rational and irrational numbers as simple continued fractions, rational approximation to irrational numbers, Hurwitz theorem, basic facts of periodic continued fractions and their illustrations (without proofs); Pell’s equation.

Textbooks: 1. An Introduction to the Theory of Numbers (3rd edition) – I. Niven and H. S. Zuckerman, Wiley Eastern Ltd., New Delhi, 1993. 2. Elementary Number Theory (4th edition) – D. M. Burton, Universal Book Stall, New Delhi, 2002.

17 Version 1.0, 2014

Reference books: 1. History of the Theory of Numbers (Vol. II, Diophantine Analysis) – L. E. Dickson, Chelsea Publishing Company, New York, 1971. 2. An Introduction to the Theory of Numbers (6th edition) – G. H. Hardy and E. M. Wright, The English Language Society and Oxford University Press, 1998. 3. An Introduction to the Theory of Numbers (6th edition) – I. Niven, H. S. Zuckerman and H. L. Montegomery, John Wiley and sons, Inc., New York, 2003.


COMPUTER PROGRAMMING Credits: 4 (2+0+2)

Computer Programming [2-0-2]

THEORY COURSE: UNIT I:

Character sets for C; constants and variables in C; arithmetic expressions in C; assignment and multiple assignments and mode of statements in C; built-in functions and libraries in C; input and output statements in C; comment statements; data types; TYPE declarations; statement labels; elementary programs in C.

UNIT II: Logical IF statements in C; switch, break, continue GOTO statements in C; WHILE, FOR, DO WHILE loops in C. Subscripted variables and arrays in C; array variables, syntax rules, use of multiple subscripts in arrays, reading and writing multi-dimensional arrays, for loops, for arrays in C; format specifications in C.

UNIT III: Some algorithms and programs on theory of matrices and numbers like Sieve method for primality test, generation of twin primes, solution of congruence using complete residue system, addition, subtraction and multiplication of matrices, transpose, and determinant .

UNIT IV: Function definition, function prototypes, arguments, call by value, call by reference, pointers, character arrays, automatic variables in C; external variables and scopes in C; some applications of C; operations with strings and sorting; file processing commands.

Textbooks: 1. Computer Programming in C – V. Rajaraman, Prentice-Hall of India Pvt. Ltd., 2005.

18 Version 1.0, 2014

2. Computer Applications of Mathematics and Statistics – A. K. Chattapadhyay and T. Chattapadhyay, Asian Books Pvt. Ltd., New Delhi, 2005. Reference books: 3. The C Programming Language – B. W. Kernighan and D. M. Ritchie, Prentice Hall, India, 1995. 4. Primes and Programming – An Introduction to Number Theory with Programming – P. Goblin, Cambridge University Press, 1993.

PRACTICALS: The following programs are to be practised:

1. Determination of roots of quadratic equations, Ax2+Bx+C=0, 2. Arranging given set of numbers in increasing/decreasing order, calculation of Mean, 3. Evaluation of sum of power series eg. ex, sin x, cos x, log (1 + x). 4. Calculation of GCD/LCM of two integers. 5. Evaluation of factorial of a positive integer and evaluation of binomial coefficients. 6. Evaluation of factorial of binomial coefficients mod 2. 7. Sieve method for primality test. 8. Generation of twin primes. 9. Solution of congruence using complete residue system. 10. Evaluation Legendre polynomial from recurrence relation. 11. Addition, subtraction and multiplication of matrices. 12. Transpose, determinant. 13. Inversion of real or complex matrices. 14. Searching a pattern in a given text and replacing every occurrence of it with another given

string. 15. Writing a given number in words using function. 16. Arranging a set of names in alphabetical order. 17. Operations with strings and sorting.


DISCRETE MATHEMATICS Credits: 4 (3+1+0)

UNIT-I: Sets and classes, Relations and functions, Equivalence relations and equivalence classes, Principle of mathematics induction, Recursive definitions, Posets, Chains and well ordered sets, Axiom of choice, Cardinal and ordinal numbers, Cantor’s lemma, Set theoretic paradoxes.

UNIT-II: Propositional Calculus: Well-formed formulas, Tautologies, Equivalence, Normal forms, Truth of algebraic systems, Calculus of predicates.

19 Version 1.0, 2014

UNIT-III: Principles of addition and multiplication, Arrangements, Permutation and combinations, Multinomial theorem, Partitions and allocations, Pigeonhole principle, Inclusion-exclusion principle, Generating functions, Recurrent relations.

UNIT-IV: Graphs and digraphs, Eulerian cycle and Hamiltonian cycle, adjacency and incidence matrices, vertex colouring, planarity and duality, trees, spanning trees, minimum spanning trees.

UNIT-V: Applications of graph theory to transport networks, matching theory and graphical algorithms, spectra of graphs, graph colorings, Ramsey theory.

Textbooks: 1. J.P. Tremblay and R.P. Manohar , Discrete Mathematics with Applications to Computer Science, McGraw Hill , 1989. 2. V. K. Balakrishnan, Introductory Discrete Mathematics, Dover, 1996 3. F. Harary, Graph Theory, Narosa, 1995 3. Bela Bollobas, Graph Theory: An Introductory Course, GTM, Springer Verlag, 1990. 4. A First Course In Theory Of Numbers: K.C.Chowdhury; Asian Books Private Ltd.

20 Version 1.0, 2014

SPECIAL PAPERS

MTH 1101C: THEORY OF RELATIVITY UNIT-I:

The special theory of relativity: inertial frames of reference; postulates of the special theory of relativity; Lorentz transformations; length contraction; time dilation; variation of mass; composition of velocities; relativistic mechanics; world events, world regions and light cone; Minkowski space-time; equivalence of mass and energy.

UNIT II: Energy-momentum tensors: the action principle; the electromagnetic theory; energy-momentum tensors (general); energy-momentum tensors (special cases); conservation laws.

UNIT III: General Theory of Relativity: introduction; principle of covariance; principle of equivalence; derivation of Einstein's equation; Newtonian approximation of Einstein's equations.

UNIT IV: Solution of Einstein's equation and tests of general relativity: Schwarzschild solution; particle and photon orbits in Schwarzschild space-time; gravitational red shift; planetary motion; bending of light; radar echo delay.

UNIT V: Brans-Dicke theory: scalar tensor theory and higher derivative gravity; KaluzaKlein theory.

Textbooks: 1. The Theory of Relativity (2nd edition) – R.K. Pathria, Hindustan Publishing co.Delhi, 1994. 2. General Relativity & Cosmology (2nd edition) – J.V. Narlikar, Macmillan co. of India Limited, 1988. Reference books: 1. Aspects of Gravitational Interactions – S. K. Srivastava and K. P. Sinha, Nova Science Publishers Inc. Commack, New York, 1998. 2. Essential Relativity – W. Rindler, Springer-Verlag, 1977. 3. General Relativity – R.M. Wald, University of Chicago Press, 1984.

MTH 1102C: FLUID DYNAMICS

UNIT-I: Waves Long wave and surface wave, stationary wave, Energy of the waves, Waves between different media, Group velocity, Dynamical significance of Group velocity, Surface tension and Capilliary waves, Effect of Surface tension in water waves.

21 Version 1.0, 2014

UNIT II: Viscous fluid motion Navier-Stokes equation of motion, rate of change of vorticity and circulation, rate of dissipation of energy, diffusion of a viscous filament.

UNIT III: Exact solution of Navier-Stokes Equation Flow between plates, Flow through a pipe (circular, elliptic), Suddenly accelerated plane wall, Flow near an Oscillating flat plate, Circular motion through cylinders.

UNIT IV: Laminar Boundary Layer Theory General outline of Boundary layer flow, Boundary layer thickness, Displacement thickness, Energy thickness, Flow along a flat plate at zero incidence, Similarity solution and Blasius solution for flow about a flat plate. Karman’s momentum integral equation, Energy integral equation, Pohlhausen solution fo momentum integral equation. Two-dimensional Boundary layer equations for flow over a curved surface, Blasius solution for flow past a cylindrical surface, phenomenon of separation.

Textbooks: 1. Hydrodynamics by Horace Lamb, CUP. 2. Theoritical Hydrodynamics by L.M.Milne Thomson, McMIllan Company. 3. Boundary Layer Theory: H. Schlichting translated by J. Kertin, McGraw Hill, New York. Reference books: 1. Modern Development of Fluid Dynamics, Vol –I by S. Goldstein, Dover Publication, New York. 2. An Introduction to Fluid Dynamics by G.K. Batchelor, Foundation Books, New Delhi. 3. Hydrodynamics by Raisinghania.

MTH 1103C: SPACE DYNAMICS UNIT-I: Basic formulae of spherical triangle – The Two Body problem

The motion of the centre of mass, the relative motion, Keplar’s equation, Solution by Hamilton-Jacobi Theory. The Determination of Orbits: Laplace;s Gauss Methods

UNIT II: The Three Body problem General Three Body problem, Restricted Three Body problem, Jacobi integral, Curves of zero velocity, Stationary solutions and their stability. The n-body problem: The motion of the centre of mass, Classical integrals.

UNIT III: Perturbation Osculating orbit, perturbing forces, Secular and Periodic perturbations, Lagrange’s planetary equations in terms of perturbing forces and in terms of aperturbed Hamiltonian. Motion of the moon – The perturbing forces, Perturbation of Keplerian elements of the moon by the sun.

22 Version 1.0, 2014

UNIT IV: Flight Mechanics Rocket performance in a vacuum, vertically ascending paths. Gravity twin trajectories, Multi-stage rocket in a vacuum. Definitions pertinent to a single stage rocket, performance, limitations of a single stage rockets. Definitions pertinent to a multi-stage rockets, Analysis of a multi-stage rockets neglecting gravity, Analysis of multi-stage rockets including gravity.

UNIT V: Rocket performance with Aerodynamic forces Short-range non-lifting missiles. Ascent of a sounding rocket, some approximate performance of rocket powered aircraft.

Textbooks: 1. Fundamentals of Celestial Mechanics by JMA Danby, The Macmillan Company, 1962. 2. Celestial Mechanics by E.Finlay, Freuhdlich, The Macmillan Company, 1958. 3. Orbital Dynamics of Space Vehicles by Ralph Deutsch, Prentice Hall Inc., New Jersey, 1963. Reference books: 1. An Introduction to Celestial Mechanics by Theodre E. Stern, 1960, Intersciences Publishers Inc. 2. Flight Mechanics Vol-I, Theory of Flight paths, by Angelo Miele, 1962, Addison Wiley Publishing Company Inc.

MTH 1104C: DYNAMICAL SYSTEMS UNIT-I:

Dynamical systems and Vector fields, the fundamental theorem, existence and Uniqueness, continuity of solution in initial condition, orbit of a map, fixed point, equilibrium point, periodic point, circular map, configuration space and phase space.

UNIT II: Stability of a fixed point, equilibrium point, concept of limit cycle and torus, hyperbolicity. Quadratic Map Period doubling phenomenon, Feigenbaum’s universal constant.

UNIT III: Nonlinear oscillators – conservative system. Hamiltonian system. Various types of Oscillators in nonlinear system. Solutions of nonlinear differential equations.

UNIT IV: Phenomenon of losing stability, Quasiperiodic motion. Topological study of nonlinear differential equations. Poincare map.

UNIT V: Randomness of orbits of a dynamical system. Chaos. Strange attractors. Various routes to chaos. Onset mechanism of turbulence.

23 Version 1.0, 2014

UNIT VI: Basic idea of Fractal geometry, construction of the middle third Cantor set, Von Koch curve, Sierpinski gasket, self similar fractals with different similarity ratio, Julia set, measure and mass distribution, Housdorff measure, Housdorff dimension and its properties.

UNIT VII: Measurement of asset at scale box dimension, its equivalent versions, properties of box dimension, box dimension of middle third Cantor set and other simple sets, upper estimate of box dimension, generalized Cantor set and its dimension.

Textbooks: 1. Robert L. Devany: An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Co. Inc, 1989. 2. Morris W. Hirsch, Stephen Smale: Differential Equation, Dynamical System and Linear Algebra, Academic Press. 3. Kenneth Falconer: Fractal Geometry, John Wiley and sons, 1995. Reference books: 1. V.I. Arnold: Dynamical systems, Bifurcation Theory and Catastrophy Theory, Springler Verlag, 1992. 2. D.K. Arrowsmith: Introduction to Dynamical Systems, CUP, 1990. 3. K.J. Falconer: The Geometry of Fractals Sets, CUP, 1985. 4. M.F. Barnsley: Fractals everywhere, A.P., 1988.

MTH 1105C: RING THEORY UNIT I:

Basic concepts of rings, modules, operations on ideals and sub-modules; matrix rings, polynomial rings; direct products of rings; fields and division rings; idempotent and nilpotent elements in a ring.

UNIT II: Isomorphism theorems; exact sequences; the group of homomorphisms and its properties relative to exact sequences.

UNIT III: Direct sums and direct products of modules, external and internal direct sums, direct summands; Zorn’s lemma, every vector space has a basis; free modules and projective modules; torsion free and torsion modules over commutative domains; exact sequences and projectivity.

UNIT IV: Injective modules, injectivity and divisibility over domains; exact sequences and injectivity; Baer's theorem and its elementary applications; simple modules, semi-simple modules (as per Bourbaki); Schur’s lemma.

24 Version 1.0, 2014

UNIT V: Equivalent conditions for semisimple modules; Wedderburn structure theorem (only statement); characterization of semisimple rings via projective and injective modules.

Textbooks: 1. Algebra, Vol. 2: Rings – I. S. Luthar and I.B.S. Passi, Narosa Publishing House, New Delhi, 1999. 2. Elementary Rings and Modules – I. T. Adamson, Oliver and Boyd, Edinburgh, 1995. 3. Basic Algebra II (3rd edition) – N. Jacobson, Hindustan Publishing Corporation, New Delhi, 2002. Reference books: 1. Algebra, Second Edition – S. Lang, Addison-Wesley, Massachusetts, 1984. 2. Notes on Homological Algebra – J. J. Rotman, Van nostrand, 1990.

MTH 1106C: HOMOLOGICAL ALGEBRA

[Preliminaries: Ring and modules, projective modules, injective modules, semi-simple rings, hereditary rings,]

I. Additive functors: definitions, operators, preservation of exactness, composite functors, change of rings,

II. Homology: modules with differentiation, the ring of dual numbers, graded modules, complexes, functors of complexes, the homomorphism α,

III. Derived functors: complexes over modules, definitions of derived functors, sums, products, limits

IV. Derived functors of ⊗ and Hom: the functor Tor and Ext, dimension of modules and rings, Kunneth relations,change of rings, duality homomorphisms.

V) The field op quotients, inversible ideals.

VI) Augmented rings: homology and cohomology of an augmented ring, examples, change of rings,dimension, faithful systems, applications to graded and local rings,

V. Algebras and tensor products, associative formulae, enveloping algebra, homology and cohomology of algebras, homology and supplemented algebras,external products, formal properties of products,isomorphisms, internal products, products for supplemented algebras,

VI. Finite groups:norms, complete derived sequences, complete solutions,products for pinite groups, uniqueness theorem, duality, examples,

VII. Extension op modules, associated algebras,supplemented algebras, groups VIII. Spectral sequences, : filtrations, convergence, maps, homotopies, graded case, induced

homomorphisms and exact sequences, applications. Books: 1. Hilton and Stammbach : A Course in Homological Algebra (Springer Graduate Texts in Mathematics)

25 Version 1.0, 2014

2. Joseph Rotman : Notes on Homological Algebra 3. Northcott : Introduction to Homological Algebra 4. MacLane : Homology 5. Cartan and Eilenberg : Homological Algebra 6. Jans : Rings and Homology. 7. Massey W.S. : Homology and Cohomology Theory (monographs and text books, in pure and applied

mathematics )vol 46,Marcel Dekker inc. 8. Massey,W.S.: Singular Homology theory,(graduate text in Mathematics-)vol-70,Springer-verlag

MTH 1107C: ALGEBRAIC TOPOLOGY

UNIT I: Homotopy of paths, fundamental group of a topological space, fundamental group functor, homotopy of maps of topological spaces; homotopy equivalence; contractible and simply connected spaces; fundamental group of S^1, S^1 x S^1 etc.; degree of maps of S^1.

UNIT II: Calculation of fundamental groups of n (n > 1) using Van Kampen's theorem (special case); fundamental group of a topological group; Brouwer's fixed point theorem; fundamental theorem of algebra; vector fields, Frobenius theorem on eigenvalues of 3 x 3 matrices.

UNIT III: Covering spaces, unique lifting theorem, path-lifting theorem, covering homotopy theorem, applications; criterion of lifting of maps in terms of fundamental groups; universal coverings and its existence; special cases of manifolds and topological groups.

UNIT IV: Simplicial and singular homology, reduced homology, Eilenberg-Steenrod axioms (without proof), relation between П1 and H1; relative homology.

UNIT V: Calculations of homology of S^n; Brouwer's fixed point theorem for f : E^n → E^n (n > 2) and its applications to spheres and vector fields; Meyer-Vietoris sequence and its application.

Textbooks: 1. Topology, a first course – J. R. Munkres, Prentice-Hall of India Ltd., New Delhi, 2000. 2. Algebraic topology, a first course (2nd edition) – M. J. Greenberg and J. R. Harper, Addison-Wesley Publishing co., 1997. 3. Algebraic Topology – A. Hatcher, Cambridge University Press, 2002. Reference books: 1. Algebraic Topology (2nd edition) – E. H. Spanier, Springer-Verlag, New York, 2000. 2. An Introduction to Algebraic Topology – J. J. Rotman, Graduate Text in Mathematics, No. 119, Springer, New york, 2004.

26 Version 1.0, 2014

3. Algebraic topology, a first course (2nd edition) – W. Fulton, Graduate Text in Mathematics, No. 153, Springer, New york, 1995. 4. Foundations of Algebraic Topology (2nd edition) – S. Eilenberg and N. E. Steenrod, Princeton University Press, 1995.

MTH 1108C: DIFFERENTIAL GEOMETRY

UNIT I: Vectors; tangent vectors; tangent spaces; tangent vector fields; derivative mappings; translations; affine transformations and rigid motions (isometries); exterior derivatives.

UNIT II: Space curves; arc length; tangent vectors and vector fields on a curve; curvature and torsion; Serret-Frenet formulas; osculating plane; osculating circle; osculating sphere; fundamental theorem of local theory of space curves (existence and uniqueness theorems).

UNIT III: Surfaces and their (local) parametrization on coordinate systems; change of parameters; parametrized surfaces; curves on surfaces; tangent and normal vectors; tangent and normal vector fields on a surface; first, second and third fundamental forms of a surface at a point; Gauss mapping.

UNIT IV: Normal sections and normal curvature of a surface at a point; Meusnier’s theorem; elliptic, hyperbolic, parabolic and planar points; Dupin indicatrix; principal directions; principal curvatures of a surface at a point; Mean curvature and Gaussian curvature of a surface at a point.

UNIT V: Line of curvature; asymptotic curves; conjugate directions; fundamental equations of the local theory of surfaces; statement of Bonnet’s fundamental theorem of local theory of surfaces.

Textbook: 1. A first course in Differential Geometry – Chun-Chin Hsiung, Willey-Interscience Publications, John Wiley & Sons, 1981. 2. An Introduction to differential geometry – T. S. Willmore, Oxford, Clarendon Press, 1979. Reference books: 1. A treatise on the differential geometry of curves and surfaces – P. Eissenhart, Dover Publications, Inc., New York, 1960. 2. Differential Geometry of three dimensions – C. R. Weatherburn, The English Language Book Society and Cambridge University Press, 1964. 3. A course in differential geometry – W. Klingenberg, Graduate Texts in Mathematics 51, Springer-Verlag, 1978. 4. Elementary differential Geometry – A. Pressley, Springer International Edition, 2005.

27 Version 1.0, 2014

MTH 1109C: INTRODUCTION TO LIE ALGEBRAS

Lie algebras and Lie algebra homomorphisms (definition and examples), solvable and nilpotent Lie, Engel's theorem; Semisimple Lie algebras - Lie's theorem, Cartan's criterion, Jordan-Chevalley decomposition, killing form, complete reducibility of representations, Weyl's theorem, irreducible representations of the Lie algebra SL(2), weights and maximal vectors, root space decomposition; Root systems - definition and examples, simple roots and the Weyl group, Cartan matrix of a root system, Dynkin diagrams, classification theorem.

Texts/References: 1. J. E. Humphreys, Introduction to Lie Algebras and Representation theory, Graduate texts in

Mathematics, Springer, 1972. 2. W. Fulton and J. Harris, Representation theory - A First Course, Graduate texts in Mathematics,

Springer, 1991. 3. K. Erdmann and M. J. Wildon, Introduction to Lie Algebras,Springer Undergraduate Mathematics

Series, 2006. 4. B.C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Graduate

Texts in Mathematics, Springer, 2010. 5. N. Jacobson, Lie Algebras, Courier Dover Publications, 1979.

MTH 1110C: GALOIS THEORY

Field extensions, algebraic extensions, minimal polynomials, separable and normal extensions; Automorphism groups, fixed fields, Galois extensions, Galois groups; Fundamental theorem of Galois theory, Galois closure; Galois groups of finite fields; Cyclotomic extensions, abelian extensions over rationals, Kronecker-Weber theorem; Galois groups of polynomials, symmetric functions, discriminant, Galois groups of quadratic, cubic and quartic polynomials; solvable extensions, radical extensions, solution of polynomial equations in radicals, insolvability of the quintic; cyclic extensions, Kummer theory, Artin-Schreier extensions; Galois groups over rationals, transcendental extensions, infinite Galois groups.

Texts/References: 1. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley & Sons, Inc., II Edition, 1999. 2. I. Stewart, Galois Theory, Academic Press, 1989. 3. J.P. Escofier, Galois Theory, Springer, 2001. 4. Emil Artin, Galois Theory, University of Notre Dame Press,1971

MTH 1111C: NUMERICAL LINEAR ALGEBRA

28 Version 1.0, 2014

Fundamentals - overview of matrix computations, norms of vectors and matrices, singular value decomposition (SVD), IEEE floating point arithmetic, analysis of roundoff errors, stability and ill-conditioning; Linear systems - LU factorization, Gaussian eliminations, Cholesky factorization, stability and sensitivity analysis; Jacobi, Gauss-Seidel and successive overrelaxation methods; Linear least-squares - Gram- Schmidt orthonormal process, rotators and reflectors, QR factorization, stability of QR factorization; QR method linear least-squares problems, normal equations, Moore- Penrose inverse, rank deficient least-squares problems, sensitivity analysis. Eigenvalues and singular values - Schur's decomposition, reduction of matrices to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; QR algorithm, implementation of implicit QR algorithm; Sensitivity analysis of eiegnvalues; Reduction of matrices to bidiagonal forms, QR algorithm for SVD.

Software Support: MATLAB.

Texts: 1. L. N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997. 2. D. S. Watkins, Fundamentals of Matrix Computation, 2nd Edn., Wiley, 2002.

References: 1. J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997. 2. B. N. Datta, Numerical Linear Algebra and Applications, 2nd Edn., SIAM, 2010. 3. G. H. Golub and C.F.Van Loan, Matrix Computation, 3rd Edn., Hindustan book agency, 2007.

MTH 1112C: FUZZY SETS AND APPLICATION UNIT I: Fuzzy sets

Basic definitions, d-level sets, convex fuzzy sets, basic operations on fuzzy sets, types of fuzzy sets, Cartesian products, algebraic products, bounded sum and difference, t-conoroms.

UNIT II: Extension principle The Zadeh extension principle, image and inverse image of fuzzy sets, fuzzy numbers, elements of fuzzy arithmetic.

UNIT III: Fuzzy relations and fuzzy graphs Fuzzy relations and fuzzy sets, composition of fuzzy relations, min-max composition and its properties, fuzzy equivalencerelations, fuzzy compatibility relations, fuzzy relations equations, fuzzy graphs, similarity relation.

29 Version 1.0, 2014

UNIT IV: Possibility theory Fuzzy measures, evidence theory, necessity measure, probability measure, possibility measure, possibility distribution, possibility theory and fuzzy sets, possibility theory and probability theory.

UNIT V: Fuzzy logic An overview of classical logic, multivalued logic, fuzzy propositions, fuzzy quantifiers, linguistic variable and hedges, inference from conditional fuzzy propositions, the compositional rule of inference, application in civil, mechanical and industrial engineering.

UNIT VI: Approximate reasoning An overview of fuzzy expert system, fuzzy implications and their selection, multiconditional approximate reasoning, the role of fuzzy relation equation.

UNIT VII: Introduction to fuzzy control Fuzzy controllers, fuzzy rule base, fuzzy inference engine, fuzzification,defuzzification and the various defuzzification methods (the centre of area, the centre of maxima and the mean of maxima methods), introduction of fuzzy Neural Network, Autometa and Dynamical systems.

UNIT VIII: Decision making in fuzzy environment Individual decision making, mult-person decision making, multicriteria decision making, multi stage decision making, fuzzy ranking methods, fuzzy linear programming, application in Medicine and Economics.

Textbook: 1. G.J.Klir and B.Yuan: Fuzzy Sets and Fuzzy Logic- Theory and Applications, Prentice Hall of India, 1995. 2. H.J. Zimmermann: Fuzzy Theory and its Application, Allied Publishers Ltd., 1991

30 Version 1.0, 2014

ELECTIVE PAPERS COURSES OPEN TO ALL DEPARTMENTS

One out of the following courses will be offered in each semester according to the availability of faculty in the department. Each Course will be of 3 credits.

1. Application of Mathematics in Finance (Paper Code: MTH 1201E) 2. Operation Research (Paper Code: MTH 1202E) 3. Scientific Computing (Paper Code: MTH 1203E) 4. Mathematical Modelling (Paper Code: MTH 1204E) 5. Biomechanics (Paper Code: MTH 1205E) 6. Number Theory and Cryptography (Paper Code: MTH 1206E) 7. Optimization Techniques (Paper Code: MTH 1207E) 8. Industrial Mathematics (Paper Code: MTH 1208E) 9. Combinatorics (Paper Code: MTH 1209E) 10. Continuum Mechanics (Paper Code: MTH 1210E)

MTH 1201E: APPLICATION OF MATHEMATICS IN FINANCE

Financial Management - An overview; Nature and Scope of Financial Management; Goals of Financial Management and main decisions of financial management; Difference between risk, speculation and gambling. Time value of Money - Interest rate and discount rate; Present value and future value- discrete case as well as continuous compounding case; Annuities and its kinds. Meaning of return; Return as Internal Rate of Return (IRR); Numerical Methods like Newton Raphson Method to calculate IRR; Measurement of returns under uncertainty situations. Meaning of risk; Difference between risk and uncertainty; Types of risk; Measurements of risk. Calculation of security and Portfolio Risk and Return-Markowitz Model; Sharpe’s Single index Model; Systematic Risk and Unsystematic Risk. Taylor series and Bond Valuation; Calculation of Duration and Convexity of bonds. Financial Derivatives — Futures, Forward, Swaps and Options; Call and Put Option; Call and Put Parity Theorem; Pricing of contingent claims through Arbitrage and Arbitrage Theorem.

References: 1. Aswath Damodaran, Corporate Finance - Theory and Practice, John Wiley 8. Sons, lnc. 2. John C. Hull, Options, Futures, and Other Derivatives, Prentice Hall of lndia Private Limited.

31 Version 1.0, 2014

3. Sheldon M. Ross, An Introduction to Mathematical Finance, Cambridge University Press. 4. Mark S. Dorfman, Introduction to Risk Management and Insurance, Prentice Hall, Englwood Cliffs, New Jersey. 5. CD. Daykin, T Pentikainen and M. Pesonen, Practical Risk Theory tor Actuaries, Chapman & Hall. MTH 1202E: OPERATION RESEARCH UNIT I: Linear Programming Problem

Introduction- Nature and Features of Operations Research (O.R)- Convex set- Polyhedral Convex Set-Linear Programming (L.P)-Mathematical Formulation of the Problem- Graphical Solution Method-Some Exceptional Cases-General Linear Programming Problem (General L.P.P) – Slack and Surplus Variables-Reformulation of the General L.P.P.- Simplex Method- Matrix Notation-Duality (Statement only of Property without Proof)- Initial Simplex Tableau- Pivot-Calculating the new Simplex TableauTerminal Simplex Tableau- Algorithm of the Simplex Method.

UNIT II: Markov Analysis Introduction- Probability Vectors-Stochastic Matrices – Regular Stochastic Matrices- Fixed Points of Square Matrices- Relationships between Fixed Points and Regular Stochastic Matrices- Markov Processes- State Transition Matrix-Transition Diagram-Brand Switching Analysis-Construction of State Transition Matrices—n-step Transition Probabilities- Stationary Distribution of Regular Markov Changes- Steady State (Equilibrium) Conditions- Markov Analysis Algorithm.

UNIT III: Games and Strategies Introduction- Two- person Zero-sum games-Pay-off Matrix – some basic terms-the Maximum –Minimal Principle-Theorem on Maximum and Minimal Values of the GameSaddle Point and Value of the Game-Rule for determining a Saddle Point-Games without Saddle Points-Mixed Strategies-Graphic solution of 2 x n and m x 2 games- Dominance Property- General rule for Dominance-Modified Dominance Property.

UNIT IV: Inventory Control Introduction- The Inventory Decisions- Costs Associated with Inventories-Factors affecting Inventory Control- Economic Order Quantity (EOQ) – Deterministic Inventory Problems with no Shortages- Case 1: The fundamental EOQ problem; Characteristics and Corollary. Case 2: EOQ Problem with Several Production Runs of Unequal Length. Case 3: EOQ Problem with Finite Replenishment (Production); Characteristics- Deterministic Inventory Problems with Shortages Case 1: EOQ Problem with Instantaneous Production and Variable Order Cycle Time; Characteristics. Case 2: EOQ Problem with Instantaneous Production of Fixed Order Cycle. Case 3: EOQ Problem with Finite Replenishment (Production); Characteristics.

Text Book: 1. Operations Research by Kanti Swarup, P.K. Gupta and Man Mohan, Published by Sultan Chand & Sons, New Delhi-110002, Ninth Edition (2002). Reference Books:

32 Version 1.0, 2014

1. Operations Research- by Friderick S. Hillier and Gerald J. Lieberman, Published by Holden-Day Inc, San Fransisco, USA. Second Edition (1974) 2. Operation Research – An Introduction by Hamdy A. Taha. Published by Prentice-Hall of India Pvt. Ltd., New Delhi- 110001, Sixth Edition (2002). MTH 1203E: Scientific Computing

Errors; Iterative methods for nonlinear equations; Polynomial interpolation, spline interpolations; Numerical integration based on interpolation, quadrature methods, Gaussian quadrature; Initial value problems for ordinary differential equations - Euler method, Runge-Kutta methods, multi-step methods, predictor-corrector method, stability and convergence analysis; Finite difference schemes for partial differential equations - Explicit and implicit schemes; Consistency, stability and convergence; Stability analysis (matrix method and von Neumann method), Lax equivalence theorem; Finite difference schemes for initial and boundary value problems (FTCS, Backward Euler and Crank-Nicolson schemes, ADI methods, Lax Wendroff method, upwind scheme).

Texts: 1. D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Ed.,

AMS, 2002. 2. G. D. Smith, Numerical Solutions of Partial Differential Equations, 3rd Ed., Calrendorn Press,

1985.

References: 1. K. E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989. 2. S. D. Conte and C. de Boor, Elementary Numerical Analysis - An Algorithmic Approach, McGraw-

Hill, 1981.

3. R. Mitchell and S. D. F. Griffiths, The Finite Difference Methods in Partial Differential Equations, Wiley, 1980.

MTH 1204E: Mathematical Modelling

The Process of Applied mathematics; Setting up first-order differential equations — Qualitative solution sketching. Difference and differential equation growth models — single-species population models. Population growth - An age structure model. The spread of Technological innovation. Higher-order linear models — A model for the detection of diabetes. Combat modes. Traffic models — Car-following models. Equilibrium speed distributions.

References: Vol. 1. Differential equation models, Eds. Martin Braun, C.S. Coleman, D.A. Drew. Vol. 2. Political and Related Models, Steven. J. Brams. W.F. Lucas, P.D. Straffin (Eds)

33 Version 1.0, 2014

Vol.3. Discrete and System models, W.F. Lucas, F.S. Roberts, Fl.M. Thrall. Vol. 4. Life Science Models, H.M. Roberts & M. Thompson. All volumes published as modules in Applied Mathematics, Springer-Verlag, 1982.

MTH 1205E: Biomechanics

Prerequisite : Fluid Mechanics (See MM 503. MM 504, MM 5'05-8) Newton’s equations of motion. Mathematical modeling. Continuum approach. Segmental Movement and Vibrations. External Flow: Fluid Dynamic Forces Acting on Moving Bodies. Flying and Swimming. Blood Flow in Heart, Lung, Arteries, and Veins. Micro-and Macrocirculation.

Recommended Text: 1. Y.C. Fung, Biomechanics, Springer-Verlag, New York lnc., 1990. MTH 1206E: NUMBER THEORY AND CRYPTOGRAPHY

Congruence, Chinese Remainder Theorem, Primitive Roots, Quadratic reciprocity, Finite fields, Arithmetic functions Primality Testing and factorization algorithms, Pseudo-primes, Fermat?s pseudo-primes, Pollard?s rho method for factorization, Continued fractions, Continued fraction method Hash Functions, Public Key cryptography, Diffie-Hellmann key exchange, Discrete logarithm-based crypto-systems, RSA crypto-system, Signature Schemes, Digital signature standard, RSA Signature schemes, Knapsack problem. Introduction to elliptic curves, Group structure, Rational points on elliptic curves, Elliptic Curve Cryptography. Applications in cryptography and factorization, Known attacks.

Texts: 1. N. Koblitz, A Course in Number Theory and Cryptography, Springer 2006. 2. I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to theory of numbers, Wiley, 2006. 3. L. C. Washington, Elliptic curves: number theory and cryptography, Chapman & Hall/CRC, 2003.

References:

1. J. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer-Verlag, 2005. 2. D. Hankerson, A. Menezes and S. Vanstone, Guide to elliptic curve cryptography, Springer-

Verlag, 2004. 3. J. Pipher, J. Hoffstein and J. H. Silverman , An Introduction to Mathematical Cryptography,

Springer-Verlag, 2008. 4. G.A. Jones and J.M. Jones, Elementary Number Theory, Springer-Verlag, 1998. 5. R.A. Mollin, An Introduction to Cryptography, Chapman & Hall, 2001.

34 Version 1.0, 2014

MTH 1207E: OPTIMIZATION TECHNIQUES

Mathematical foundations and basic definitions: concepts from linear algebra, geometry, and multivariable calculus. Linear optimization: formulation and geometrical ideas of linear programming problems, simplex method, revised simplex method, duality, sensitivity snalysis, transportation and assignment problems. Nonlinear optimization: basic theory, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, convex optimization. Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods.

Texts: 1. N. S. Kambo, Mathematical Programming Techniques, East West Press, 1997. 2. E.K.P. Chong and S.H. Zak, An Introduction to Optimization, 2nd Ed., Wiley, 2010.

References: 1. R. Fletcher, Practical Methods of Optimization, 2nd Ed., John Wiley, 2009. 2. D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd Ed., Springer India, 2010. 3. M. S. Bazarra, J.J. Jarvis, and H.D. Sherali, Linear Programming and Network Flows, 4th Ed., 2010.

(3nd ed. Wiley India 2008). 4. U. Faigle, W. Kern, and G. Still, Algorithmic Principles of Mathematical Programming, Kluwe,

2002. 5. D.P. Bertsekas, Nonlinear Programming, 2nd Ed., Athena Scientific, 1999. 6. M. S. Bazarra, H.D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms,

3rd Ed., Wiley, 2006. (2nd Edn., Wiley India, 2004).

MTH 1208E: INDUSTRIAL MATHEMATICS

UNIT I: An introduction to Financial Derivatives, types of financial derivatives – Forward and Futures; Options and its kinds and SWAPS.

UNIT II: The arbitrage theorem and introduction to Portfolio Selection and Capital Market Theory, Static and Continuous- Time model.

UNIT III: A single period option pricing model, multi period option pricing model, Cox-Ross-Rubistein model, bounds on option price.

UNIT IV: The Ito’s Lemma and the Ito’s integral.

Textbooks: 1. John C.Hull, Options, Futures and Other Derivatives, Prentice Hall of India Pvt. Ltd.

35 Version 1.0, 2014

References: 1. Sheldon M. Ross, An Introduction to Mathematical Finance, CUP. 2. Salih N. Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic Press.

MTH 1209E: GRAPH AND COMBINATORICS

Counting principles, multinomial theorem, set partitions and Striling numbers of the second kind, permutations and Stirling numbers of the first kind, number partitions, Lattice paths, Gaussian coefficients, Aztec diamonds, formal series, infinite sums and products, infinite matrices, inversion of sequences, probability generating functions, generating functions, evaluating sums, the exponential formula, more on number partitions and infinite products, Ramanujan's formula, hypergeometric sums, summation by elimination, infinite sums and closed forms, recurrence for hypergeometric sums, hypergeometric series, Sieve methods, inclusion-exclusion, Mobius inversion, involution principle, Gessel-Viennot lemma, Tutte matrix-tree theorem, enumeration and patterns, Polya-Redfield theorem, cycle index, symmetries on N and R, polyominoes

Texts: 1. M. Aigner. A Course in Enumeration. Springer, GTM, 2007.

References: 1. C. Berge. Principles of Combinatorics. Academic Press, 1971. 2. J. Riordan. Introduction to Combinatorial Analysis. Dover, 2002. 3. M. Bona. Introduction to Enumerative Combinatorics. Tata McGraw Hill, 2007.

MTH 1210E: CONTINUUM MECHANICS UNIT I: Analysis of Stress

The continuum concept, homogeneity, isotropy, mass density, Cauchy’s stress principle, stress tensor, equations of equilibrium, stress quadric of Cauchy, Principal stresses, stress invariants, deviator and spherical stress tensors.

UNIT II: Analysis of Strain Lagrangian and Eularian descriptions, deformation tensors, finite strain tensor, small deformation theory, linear strain tensors and physical interpretation, stress ratio and finite strain interpretation, strain quadric of Cauchy, Principal strains, strain invariants, spherical and deviator strain components, equations of compatibility.

UNIT III: Motion Material derivatives, path lines and stream lines, rate of deformation and vorticity with their physical interpretation, Material derivatives of volume, surface and line elements, Volume surface and line integrals, fundamental laws of continuum mechanics.

36 Version 1.0, 2014

UNIT IV: Constitutive equations of Continuum Mechanics Linear elasticity, Generalized Hook’s Law, Strain energy function, elastic constant for isotropic, homogeneous materials, elastostatic and elastodynamic problems. Fluids: Viscous stress tensor, Barotropic flow, Stokesian fluids, Newtonian fluids, Navier Stokes equations, irrotational flow, perfect fluids, Bemouli’s equation, circulation.

Text Books: 1. Continuum Mechanics by G.E.Mase, Schaum’s outline series, McGraw Hill. 2. Textbook of Fluid Dynamics by Frank Chorlton, CBS Publishers, Delhi. References: 1. Mathematical Theory of Continuum Mechanics by R. Chatterjee, Narosa Publishing House, New Delhi. 2. An Introdution to Fluid Mechanics by G.K. Batchelor, Foundation Books, New Delhi.

cotton college state universityccsu.ac.in/admin/gallery/math pg_syllabus_ccsu_july28_r.pdf ·...

Documents