B. Sc. (Hons) Mathematics Course Structure & Syllabus

Year FIRST SEMESTER SECOND SEMESTER

Course Code Course Name L T P C Course Code Course Name L T P C

I

MA1103 Calculus 3 1 0 4 MA1203 Differential Equations 3 1 0 4

MA1104 Discrete Mathematics Structure 3 1 0 4 MA1204 Number Theory 3 1 0 4

MA1105 Higher Trigonometry 3 1 0 4 MA1205 Abstract Algebra 3 1 0 4

CA1170 Fundamentals of Computer 1 1 0 2 MA1206 Three-Dimensional Geometry 3 1 0 4

CA1175 Fundamentals of Computer lab 0 0 2 1 ****** GE-II (A) 2 1 0 3

CY1003 Environmental Science 2 1 0 3 ****** GE-II (A) Lab - - 2 1

LN1106 Communicative English 1 0 2 2 ****** GE-II (B) 2 1 0 3

****** GE – I 2 1 0 3 ****** GE-II (B) Lab - - 2 1

****** GE – I Lab - - 2 1

15 6 6 24 16 6 4 24

Total Contact Hours (L + T + P) 27 Total Contact Hours (L + T + P) 26

II

THIRD SEMESTER FOURTH SEMESTER

MA2112 Real Analysis 3 1 0 4 MA2211 Multivariate Calculus 3 1 0 4

MA2113 Ring & Field Theory 3 1 0 4 MA2212 PDE & System of ODE 3 1 0 4

MA2114 Linear Programming Problems 3 1 0 4 MA2213 Linear Algebra 3 1 0 4

MA2115 Data Analysis using R 2 0 0 2 MA2214 Vector Calculus & Statics 3 1 0 4

MA2116 Introduction to C Language 2 0 0 2 MA2215 Mathematical Modeling 3 1 0 4

****** GE – III (A) 2 1 0 3 ****** DSE – I 3 1 0 4

****** GE – III (A) Lab 0 0 2 1 ****** Open Elective 2 1 0 3

****** GE – III (B) 2 1 0 3

****** GE – III (B) Lab 0 0 2 1

17 5 4 24 20 7 0 27

Total Contact Hours (L + T + P) 26 Total Contact Hours (L + T + P) + OE 24+3=27

III FIFTH SEMESTER SIXTH SEMESTER

MA3101 Numerical Analysis 2 1 0 3 MA3201 Complex Analysis 3 1 0 4

MA3102 Operations Research 2 1 0 3 MA3202 Riemann Integration & Series of Functions 3 1 0 4

MA3103 Dynamics 3 1 0 4 MA3203 Metric Space 3 1 0 4

MA3130 Lab on Numerical Analysis 0 0 2 1 ****** DSE – IV (A) 2 1 0 3

MA3131 Lab on Operation Research 0 0 2 1 ****** DSE – IV (B) 2 1 0 3

MA3170 DSE – II - Project 0 0 0 6 ****** GE – IV 2 1 0 3

****** DSE – III (A) 2 1 0 3 ****** GE – IV Lab 0 0 2 1

****** DSE – III (B) 2 1 0 3 ****** Open Elective 2 1 0 3

11 5 4 24 17 7 2 25

Total Contact Hours (L + T + P) 20 Total Contact Hours (L + T + P) + OE 23 +3 = 26

Discipline Specific Electives (DSE) DSE – I

1. MA2240 Statistical Inference# 2. MA2230 Lab on Statistical Inference# 3. MA2241 Inventory Theory and Dynamic

Programming 4. MA2242 Information Theory 5. MA2243 Mathematical Finance

DSE – III (A)

1. MA3140 Basic Econometrics 2. MA3141 Modeling & Simulation

DSE – III (B)

1. MA3142 Reliability Modeling & Analysis 2. MA3143 Portfolio Optimization

DSE – IV (A)

1. MA3241 Multivariate Analysis 2. MA3242 Numerical Methods of Ordinary

and Partial Differential Equations DSE – IV (B)

1. MA3243 Mechanics 2. MA3244 Introduction to Graph Theory

Skill Enhancement Courses (SEC) 1. CA1170 Fundamentals of Computer 2. CA1175 Fundamentals of Computer Lab 3. MA2115 Data Analysis using R 4. MA2116 Introduction to C Language

Generic Electives (GE) GE – I & Lab 1. MA1140 Descriptive Statistics 2. MA1130 Lab on Descriptive Statistics 3. CY1160 General Chemistry-I 4. CY1138 Organic Chemistry Laboratory 5. PY1160 Mechanics and STR 6. PY1136 General Physics Lab-I

GE – II (A) & Lab 1. MA1240 Probability Theory & Random Variables 2. MA1230 Lab on Probability & Random Variables 3. CY1260 General Chemistry-II 4. CY1238 Inorganic Chemistry Laboratory 5. PY1261 Electromagnetism

GE – II (B) & Lab 1. MA1241 Applied Statistics 2. MA1231 Lab on Applied Statistics 3. CY1261 General Chemistry-III 4. CY1239 Physical Chemistry Laboratory 5. PY1260 Oscillation and Wave Optics 6. PY1236 Optics Lab

GE – III (A) & Lab 1. MA2140 Distribution Theory 2. MA2130 Lab on Distribution Theory 3. CY2160 Analytical Chemistry 4. CY2138 Analytical Chemistry Laboratory 5. PY2161 Heat and Thermodynamics

GE – III (B) & Lab 1. MA2141 Sampling Theory 2. MA2131 Lab on Sampling Theory 3. CY2161 Structure of Materials 4. CY2139 Material Chemistry Laboratory 5. PY2160 Electronics

GE – IV & Lab 1. MA3240 Design of Experiment 2. MA3230 Lab on Design of Experiment 3. CY3260 Biophysical Chemistry 4. CY3238 Applied Chemistry Laboratory 5. PY3260 Modern Physics

Open Electives (OE) 1. MA2080: Introduction to Astronomical

Science 2. MA2081: Regression Analysis and

Forecasting 3. MA2082: Biostatistics 4. MA3080: Marketing Research and

Analysis 5. MA3081: Mathematical Finance 6. MA3082: Modelling & Simulation 7. MA3083: Optimization Techniques

# - Papers MA2240 and MA2230 can be opted together only, total credits remain same as mentioned for the DSE-I

MA1103: CALCULUS [3 1 0 4] Limits, Continuity and Mean Value Theorem: Definition of limit and continuity, types of discontinuities, properties of continuous functions on a closed interval, differentiability, Rolle’s theorem, Lagrange’s and Cauchy’s first mean value theorems, Taylor’s theorem (Lagrange’s form), Maclaurin’s theorem and expansions, convexity, concavity and curvature of plane curves, Formula for radius of curvature in Cartesian, parametric, polar and pedal Forms, Centre of curvature, evolutes and involutes, envelopes, asymptotes, singular points, cusp, node and conjugate points, tracing of standard Cartesian, polar and parametric curves; Partial Differentiation: First and higher order derivatives, Euler’s theorem, total derivative, differentiation of implicit functions and composite functions, Taylor’s theorem for functions of two variables; Maxima & Minima: Maxima-minima for functions of two variables, necessary and sufficient condition for extreme points, Lagrange multipliers; Integral Calculus: Reduction formulae, application of integral calculus, length of arcs, surface areas and volumes of solids of revolutions for standard curves in Cartesian and polar forms; Beta and Gamma Functions: Beta and Gamma functions and relation between them, evaluation of integrals using Beta and Gamma functions. References:

1. S. Narayan and P. K. Mittal, Differential Calculus, S. Chand Publication, New Delhi, 2011.

2. P. Saxena, Differential Calculus, Tata McGraw Hill, New Delhi, 2014.

3. D. V. Widder, Calculus, PHI publication, New Delhi, 2012. 4. S. Narayanan, T. K. Manicavachagom and Pillay, Calculus I & II, S. Viswanathan Pvt. Ltd., Chennai, 2010. 5. M. J. Strauss, G. L. Bradley and K. J. Smith, Calculus, Dorling Kindersley Pvt. Ltd., New Delhi, 2007.

MA1104: DISCRETE MATHEMATICS STRUCTURE [3 1 0 4] Set Theory: Definition of sets, Venn diagrams, complements, Cartesian products, power sets, counting principle, cardinality and countability, proofs of some general identities on sets, pigeonhole principle; Relation: Definition, types of relation, composition of relations, domain and range of a relation, pictorial representation of relation, properties of relation, partial ordering relation; Algebraic Structure: Binary composition and its properties definition of algebraic structure, semi group, monoid, abelian group, properties of groups, permutation groups, sub group, cyclic group; Propositional Logic: Propositional logic, applications of propositional logic, propositional equivalences, topologies and contradiction, CNF and DNF, predicates and quantifiers; Combinatories: Basics of counting, permutations, combinations, inclusion-exclusion, recurrence relations (nth order recurrence relation with constant coefficients, homogeneous recurrence relations, Inhomogeneous recurrence relation), generating function (closed form expression, properties of generating function, solution of recurrence relation using generating function; Graph Theory: Graph terminology and special types of graphs, representing graphs and graph isomorphism, connectivity, Euler and Hamilton paths, planar graphs. References:

1. K. H., Rosen, Discrete Mathematics and Its Applications, 7th edition, USA, Tata McGraw-Hill, 2007. 2. J. P. Chauhan, Discrete Structures & Graph Theory, Krishna Publication, 2018. 3. V. Krishnamurthy, Combinatories: Theory and Applications, East-West Press, New Delhi, India, 2018. 4. S. Lipschutz, M. Lipson, Discrete Mathematics Tata Mac Graw Hill, 2005. 5. Kolman, Busby Ross, Discrete Mathematical Structures, Pearson Education India, 2015.

MA1105: HIGHER TRIGONOMETRY [3 1 0 4] Complex Numbers: Introduction of complex numbers, properties of complex numbers, geometrical representation of a complex number, geometrical interpretation of complex numbers; De Moivre's Theorem: Statement and proof of De Moivre's theorem for integral indices, alternative method, Proof for rational indices, all possible values of

( )/

cos sinp q

x i x+ , application of De-Moivre's theorem for integral and fractional indices; Trigonometric Functions:

Circular trigonometric functions, trigonometric functions of related angles, properties of trigonometric functions, trigonometric identities, inverse functions, summation of series; Hyperbolic and Exponential Functions: Definitions of exponential and hyperbolic functions, laws of exponential and hyperbolic functions; Inverse functions, Summation of series; Logarithm: Definition, properties of logarithms, change of base, two systems of logarithms, use of logarithmic table, antilogarithms, Exponential and logarithmic series. References:

1. R. Mazumdar, A. Dasgupta and S. B. Prasad, Degree Level Trigonometry, Bharti Bhawan, Patna, 2012. 2. Lalji Prasad, Higher Trigonometry, Paramount publications, Patna, 2016. 3. V. K. Parashar, Applied Mathematics, Galgotia Publications pvt. Ltd, 2005. 4. S. L. Loney, Plane Trigonometry, University of Michigan Library, 2005. 5. R. K. Ghosh and K. C. Maity, Higher Algebra, New Central Book Agency, Kolkata, 2013. 6. T. Veerarajan and T. Ramachandran, Numerical Methods, Tata McGraw Hill, New Delhi, 2009. 7. W. G. Kelley and A. C. Peterson, Difference equations an introduction with applications, 2nd edition, Harcourt

Academic Press, USA, 2001. CA1170 FUNDAMENTALS OF COMPUTERS [1 1 0 2]

Computer Fundamentals, Definition and Purpose, Data, Information and Knowledge, Characteristics of Computers, Classification of Computers, Generations of Computer, Basic organization of Computer, System Software and Application Software. Operating Systems and Multimedia, Types of Operating System, Windows v/s Linux, Mobile based OS, Multimedia, Definition and Types , Multimedia Software, Computer Networks, Applications of Networking, Network Topologies- Mesh, Bus, Star, Ring, Types of Network (LAN, MAN, WAN), Network Cables- Optical Fiber, Twisted, Co-axial, Network Devices- Hubs, Switch, Router, Network Interface Card, Ethernet, Internet, Introduction and Usage of Internet, Internet Connectivity Options (Wired and Wireless), IP Addressing and DNS, Website, URL, HTML, Web Browser and Search Engines, Operational Guideline of Computer Usage, Do’s and Don’ts of Computer, E-mails, Email Etiquettes, Cyber Security, Internet Frauds, Secure Password Formation , Computer Security, Malware, Virus, Ransomware, Social Media and its Impact. References:

1. R. Thareja, Fundamental of Computer, (1e) Oxford Publications, 2014. 2. K. Atul, Information Technology, (3e) Tata McGraw Hill Publication, 2008.

CA1175 FUNDAMENTALS OF COMPUTERS LAB [0 0 2 1] Computer Peripheral and Windows operations, MS WORD- Creating and formatting of a document, Introduction of cut, copy and paste operations, to explore various page layout and printing options, creating. Formatting, editing Table in MS word, Introduction of Graphics and print options in MS word, Introduce the student with mail merge option. MS EXCEL- creation of spreadsheet and usage of excel, Formatting and editing in worksheet, Sorting, Searching in Excel sheets, using formula and filter in MS excel, printing and additional features of worksheet, maintaining multiple worksheet and creating graphics chart. MS POWER POINT – creation of presentation, Power point views, creating slides and other operations, Using design, animation, and transition in slides, Internet Tools, Using Email and Outlook facilities, Google Drive, Google Forms, Google Spreadsheet, Google groups. References:

1. R. Thareja, Fundamental of Computer, (1e) Oxford Publications, 2014. 2. K. Atul, Information Technology, (3e) Tata McGraw Hill Publication, 2008.

CY1003: ENVIRONMENTAL SCIENCE [3 0 0 3] Introduction: Multidisciplinary nature, scope and importance, sustainability and sustainable development. Ecosystems: Concept, structure and function, energy flow, food chain, food webs and ecological succession, examples. Natural Resources (Renewable and Non-renewable Resources): Land resources and land use change, Land degradation, soil erosion and desertification, deforestation. Water: Use and over-exploitation, floods, droughts, conflicts. Energy resources: Renewable and non- renewable energy sources, alternate energy sources, growing energy needs, case studies. Biodiversity and Conservation: Levels, biogeographic zones, biodiversity patterns and hot spots, India as a mega‐biodiversity nation; Endangered and endemic species, threats, conservation, biodiversity services. Environmental Pollution: Type, causes, effects, and controls of Air, Water, Soil and Noise pollution, nuclear hazards and human health risks, fireworks, solid waste management, case studies. Environmental Policies and Practices: Climate change, global warming, ozone layer depletion, acid rain, environment laws, environmental protection acts, international agreements, nature reserves, tribal populations and rights, human wildlife conflicts in Indian context. Human Communities and the Environment: Human population growth, human health and welfare, resettlement and rehabilitation, case studies, disaster management, environmental ethics, environmental communication and public awareness, case studies. Field Work and visit. References:

1. R. Rajagopalan, Environmental Studies: From Crisis to Cure, Oxford University Press, 2016.

2. A. K. De, Environmental Studies, New Age International Publishers, New Delhi, 2007.

3. E. Bharucha, Text book of Environmental Studies for undergraduate courses, Universities Press, Hyderabad,

2013.

4. R. Carson, Silent Spring, Houghton Mifflin Harcourt, 2002.

5. M. Gadgil & R. Guha, This Fissured Land: An Ecological History of India, University of California Press,

1993.

6. M. J. Groom, K. Meffe Gary and C. R. Carroll, Principles of Conservation Biology, OUP, USA, 2005.

LN1106: COMMUNICATIVE ENGLISH [2 0 0 2] Communication- Definition, Process, Types, Flow, Modes, Barriers; Types of Sentences; Modal Auxiliaries; Tenses and its Usage; Voice; Reported Speech; Articles; Subject-Verb Agreement; Spotting Errors; Synonyms and Antonyms; One Word Substitution; Reading Comprehension; Précis Writing; Essay Writing; Formal Letter Writing; Email Etiquettes; Résumé & Curriculum Vitae; Statement of Purpose; Presentations References:

1. Collins English Usage. Harpers Collins, 2012. 2. Hobson, Archie Ed. The Oxford Dictionary of Difficult Words. Oxford, 2004. 3. Jones, Daniel. English Pronouncing Dictionary. ELBS, 2011. 4. Krishnaswamy, N. Modern English: A Book of Grammar Usage and Composition, Macmillan India, 2015. 5. Longman Dictionary of Contemporary English. Pearson, 2008. 6. McCarthy, M. English Idioms in Use. Cambridge UP, 2002. 7. Mishra, S. and C. Muralikrishna. Communication Skills for Engineers. Pearson, 2004. 8. Oxford Dictionary of English. Oxford UP, 2012. 9. Turton, N. D. and J.B. Heaton. Longman Dictionary of Common Errors. Pearson, 2004.

MA1203: DIFFERENTIAL EQUATIONS [3 1 0 4] Ordinary Differential Equations: Introduction, order and degree of a differential equation, formation of differential equations, general, particular and singular solution, Wronskian, its properties and applications; Equations of First Order and First Degree: Separation of variables method, homogeneous equations, equations reducible to homogeneous form, linear equations and equations reducible to linear form, exact equations, equations reducible to exact form, orthogonal trajectories in Cartesian coordinates, applications of first order equations; Equations of First Order and Higher Degree: Equations solvable for x, y and p, Clairaut’s and Lagrange’s equation, equations reducible to Claret’s form, Singular solution; Higher Order Linear Differential Equations: Higher order linear differential equations with constant coefficients and variable coefficients, simultaneous ordinary differential equations; Partial Differential Equations: definition, order and degree, formation of partial differential equations, Lagrange's method of solution, standard forms, Charpit Method. References:

1. J. L. Bansal, S. L. Bhargava and S. M. Agarwal, Differential Equations, Jaipur Publishing House, Jaipur, 2012. 2. M. D. Raisinghania, Ordinary and Partial Differential Equations, S. Chand & Comp., New Delhi, 2013. 3. S. L. Ross, Differential Equations, Wiley India, New Delhi, 2013. 4. E.A. Coddington, An Introduction to Ordinary Differential Equations, PHI Publication, New Delhi, 2011. 5. R. K. Jain and S.R.K. Iyengar, Advanced Engineering Mathematics, 4th edition, Narosa Publishing House,

2014. 6. G. F. Simmons, Differential Equations, Tata McGraw-Hill, 2006.

MA1204: NUMBER THEORY [3 1 0 4] 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 Möbius 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 , Fermat’s last theorem.

References: 1. Shirali and Yog, Number Theory, Orient Blackswan Private Limited, New Delhi, 2003. 2. N. Robinns, Beginning Number Theory, Narosa Publishing House Pvt. Limited, Delhi, 2007. 3. D. M. Burton, Elementary Number Theory, Tata McGraw-Hill Edition, Indian reprint, 2007. 4. G. E. Andrew, Number Theory, Revised Edition, Dover Publications, 2012.

MA1205: ABSTRACT ALGEBRA [3 1 0 4] Group Theory: Binary operation on a set, algebraic structure, definition of a group, abelian group, finite and infinite groups, order of a group, properties of groups, addition modulo m, multiplication modulo p, residue classes of the set of integers; Permutations: Groups of permutations, cyclic permutation, even and odd permutations, integral powers of an element of a group, order of an element of a group; Subgroups: Intersection of subgroups, cosets, Lagrange’s theorem, Euler’s theorem, Fermat’s theorem, order of the product of two subgroups of finite order, Cayley’s theorem, cyclic groups, subgroup generated by a subset of a group, generating system of group; Normal Subgroups: Conjugate elements, characteristics subgroup normalizer of an element of a group, class equation of a group, centre of a group, conjugate subgroups, invariant subgroups, quotient groups; Isomorphism and Homomorphism of Groups: Kernel of a homomorphism; fundamental theorem on homomorphism of groups, automorphisms of a group, inner automorphisms, results on group homomorphism, maximal subgroups. References:

1. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, 1994, reprint 2009.

2. N. S. Gopalakrishanan, University Algebra, New Age International (P) Ltd., 3rd edition, 2015. 3. Vijay K Khanna and S K Bhambri, A Course in Abstract Algebra, 4th edition, Vikas Publication House PVT Ltd,

2013. 4. J.B. Fraleigh, A first Course in Abstract Algebra, Pearson Education Limited, 2013. 5. I. N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi, 2013. 6. J. A. Gallian, Contemporary Abstract Algebra, Cengage learning, 2013.

MA1206: THREE DIMENSIONAL GEOMETRY [3 1 0 4] Line and Plane: Direction cosines of a line, direction ratios of the join of two points, projection on a line, angle between the lines, equation of line in different forms, equation of a plane in different forms, angle between two planes, line of intersection of two planes, angle between a line and a Plane; Sphere: Definition, equation of a sphere, general equation of a sphere, great circle, edquation of circle, tangent line and tangent plane of a sphere, condition of tangency for a line and equation of tangent plane, angle of intersection of two spheres, condition of orthogonality of two spheres; Cone: Cone, quadratic cone, equation of a cone, enveloping cone, condition for general equation of second degree to represent a cone, intersection with a line, tangent plane, reciprocal cone, right circular cone; Cylinder: Definition, equation of a cylinder, enveloping cylinder, equation of enveloping cylinder, right circular cylinder, equation of right

2 2 2x y z+ =

circular cylinder; Central Conicoids: Conicoids, central conicoid, standard equation of ellipsoid, hyperboloid of one sheet and hyperboloid of two sheets, nature and shape of central conicoids, tangent line and tangent planes, condition of tangency. References:

1. S. L. Loney, The Elements of Coordinate Geometry, Macmillan and Co., London, 2001. 2. P. K. Jain and Khalil Ahmad, A text book of Analytical Geometry of Three Dimensions, Wiley Eastern Ltd, 2008. 3. R. J. T. Bell, Elementary Treatise on Coordinate Geometry of Three Dimensions, Macmillan India Ltd, 1998. 4. N. Saran and R. S. Gupta, Analytical Geometry of Three Dimensions, Pothisala Pvt. Ltd, Allahabad, 2001. 5. Gorakh Prasad and H. C. Gupta, Text book on Coordinate Geometry, Pothisala Pvt. Ltd., Allahabad, 2004. 6. Sharma & Jain, Co-ordinate Geometry, Galgotia Publication, Dariyaganj, New Delhi, 1998. 7. Shanthi Narayan, Analytical Solid Geometry, New Delhi: S. Chand and Co. Pvt. Ltd., 2004.

MA2112: REAL ANALYSIS [3 1 0 4] Real Numbers: Field structure and order structure, order properties of R and Q, characterization of interval, bounded and unbounded sets, supremum and infimum, order completeness property, archimedean property, density of rational numbers in R, density theorem, characterization of intervals, absolute value of a real number, neighborhoods, open sets, closed sets, limit points of a set, Bolzano-Weierstrass theorem, isolated points, closure, nested interval, cantor nested interval theorem, cover of a set, compact set, Heine-Borel theorem, idea of countable sets, uncountable sets and uncountability of R; Real Sequences: Sequences, bounded sequences, convergence of sequences, limit point of a sequence, Bolzano-Weierstrass theorem for sequences, limits superior and limits inferior, Cauchy’s general principle of convergence, Cauchy sequences and their convergence criterion; Algebra of Sequences: Cauchy’s first and second theorems and other related theorems, monotonic sequences, subsequences; Infinite Series: Definition of infinite series, Sequence of partial sums, convergence and divergence of infinite series, Cauchy’s general principle of convergence for series, positive term series, geometric series, comparison series; Comparison Tests: Cauchy’s n th root test; Ratio test, Raabe’s test, Logarithmic test, Cauchy’s Integral test, Gauss test, alternating series and Leibnitz's theorem, absolute and conditional convergence. References:

1. S. C. Malik and S. Arora, Mathematical Analysis, New Age Int. Pub., New Delhi, 2017. 2. Shanti Narayan, Elements of Real Analysis, S. Chand & Co., New Delhi, 2015. 3. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd edition, John Wiley & Sons, 2011. 4. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw Hill, New York, 2013. 5. H. L. Royden and P. M. Fitzpatrick, Real Analysis, 3rd edition, Macmillan, New York, 2010. 6. T. M. Apostal, Mathematical Analysis, Addison-Wesley, 2008. 7. R. R. Goldberg, Methods of Real Analysis, John Wiley & Sons, 2012.

MA2113: RING AND FIELD THEORY [3 1 0 4] Rings: Zero divisors, commutative ring with identity, integral domains, division rings, subrings and ideals, congruence modulo a subring relation in a ring, simple ring, algebra of ideals, ideal generated by a subset, quotient rings, prime and maximal ideals, homomorphism in rings, natural homomorphism, kernel of a homomorphism, fundamental theorem of homomorphism, first and second isomorphism theorems, field of quotients, embedding of rings, ring of endomorphisms of an abelian group; Factorization in Integral Domains: Prime and irreducible elements, H.C.F. and L.C.M. of two elements of a ring, principal ideals domains, euclidean domains, unique factorisation domains, polynomials rings, algebraic and transcendental elements over a ring, Factorization in polynomial ring R[x], division algorithm in R[x] where R is a commutative; Ring With Identity: Properties of polynomial ring R[x] if R is a field or a U.F.D., Gauss lemma, Gauss Theorem and related examples; Field: Field extensions, finite field extensions, finitely generated extensions of a field, simple extension of a field, algebraic extension of a field, splitting (Decomposition) fields, multiple roots, normal and separable extension of a field. References:

1. S. Singh, Q. Zameeruddin, Modern Algebra, Vikas Pub. House Pvt Limited, 2009. 2. J.B. Fraleigh, A first Course in Abstract Algebra, Pearson Education Limited, 2013. 3. J. A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage Learning, USA, 2010. 4. I.T. Adamson, Introduction to Field Theory, New edition, Cambridge University Press; 2012.

MA2114: LINEAR PROGRAMMING PROBLEMS [3 1 0 4] Linear Programming Problems (LPP): Introduction, formulation of an LPP, Graphical method of solution of LPP, Areas of application of linear programming; Optimal Solution: Definitions, convex combination and convex set, extreme point, convex hull and convex polyhedron; Simplex Method: Fundamental theorem of linear programming, reduction of a feasible solution to a basic feasible solution, optimality condition, unboundedness, simplex algorithm, simplex method for maximization case of an LPP, minimization case- Big M method, Two phase method; Duality: concept of duality, mathematical formulation of duals-construction of duals, duality and simplex method; Dual Simplex Method: Introduction, dual simplex method, computational procedure of the dual simplex algorithm, initial basic solution; Transportation Problem: Introduction, mathematical formulation, initial solution by North West corner rule, Least Cost Method and Vogel’s approximation method (VAM), MODI’s method for testing optimality, special cases of transportation problem; Assignment Problem: Mathematical formulation, Hungarian method to find optimal assignment, unbalanced assignment problem.

References: 1. J. G. Chakraborty and P. R. Ghosh, Linear Programming and Game Theory, Maulik Library, Kolkata, 2010. 2. J. K. Sharma, Operations Research, Macmilan Pub. India Ltd., 2013. 3. Kanti Swarup, Gupta, P.K. and Manmohan, Operations Research, 13th edition, Sultan Chand and Sons, 2007. 4. H. A. Taha, Operations Research – An Introduction, 6th edition, Prentice Hall of India, New Delhi, India, 2000. 5. V. K. Kapoor, Operations Research, Sultan Chand & Sons., New Delhi, India, 2005. 6. S. D. Sharma, Operations Research, Kedarnath Ramnath, Meerut, 2018. 7. G. Hadley, Linear Programming, Narosa Publications, 2002.

MA2115: DATA ANALYSIS USING R [2 0 0 2] General Introduction into the R Ecosystem: Downloading and installing R, History of R, R packages, CRAN. Introduction to RStudio, Vector, Matrices and Arrays, Factors and Data Frames, Lists, Conditional and Control Flow; R Syntax Basics: Constants, operators, functions, variables, Random numbers, Vectors and vector indexing, simple descriptive stats, Loops, Conditional expressions; Data Management: Creating, recoding renaming variables, missing values, sorting, merging. Data Interface, CSV files, Excel files; Charts and Graphs: Introduction, Pie Chart, Bar Chart, Box Plot, Histogram, Line Graph and Scatter Plot. References:

1. M. Gardener: Beginning R: The Statistical Programming Language, Wiley Publications, 2012. 2. W. J. Braun and D. J. Murdoch: A First Course in Statistical Programming with R. Cambridge University Press.

New York, 2007. 3. K.G. Srinivasa and G. M. Siddesh: Statistical Programming in R, Oxford University Press, New Delhi, 2017. 4. Peter Dalgaard: Introductory Statistics with R. Springer, 2nd edition, 2008. 5. Phil Spector: Data Manipulation with R. Springer, New York, 2008. 6. Alain F. Zuur, Elena N. Ieno, and Erik Meesters: A Beginner’s Guide to R. Use R. Springer, 2009. 7. John Verzani, Using R for Introductory Statistics, Chapman & Hall/CRC, 2004.

MA2116: INTRODUCTION TO C LANGUAGE [2 0 0 2] C-Language Preliminaries: General introduction of computers, hardware and software, computer language and programming, introduction of algorithm method, introduction of flow charts, character set, keywords, character constants, ‘C’ variables, naming the variable, types of variables, declaring variable; Operator and Expressions: Operators , expressions, operators precedence in expressions; Input and Output in C-Programs: Formatted input functions, formatted output functions, unformatted input functions, unformatted output functions, mathematical library; Statements: Conditional statement, compound Statement, if statement, if-else statement; Implementing Loops in C-Programs: Loop, while statement, for loop, nesting of loops, do-while loop; Array Variables and Functions: Defining an array in C-language, multidimensional array, initializing two-dimensional array, sorting of arrays, syntax rules for function declaration. References:

1. E. Balaguruswamy, Computing Fundamentals & C Programming, Tata McGraw Hill, 2008. 2. Y. P. Kanetkar, Let us C, 12th Edition, BPB Publication, 2014. 3. C.S. Lipschutz, Programming in C, Tata McGraw Hill, 2003. 4. B. A. Forouzan & R. F. Gilberg, Computer Science – A structured programming Approach Using C, Cengage

Learning, 2011. 5. E. Balaguruswamy, Programming in ANSI-C, Tata McGrawHill, 2011. 6. B. W. Kernighan, D. M. Ritchie, The C Programing Language, 2nd edition, Prentice Hall of India, 2014.

MA2211: MULTIVARIATE CALCULUS [3 1 0 4] Limit, Continuity of Function of Several Variables and Partial Derivatives: Introduction to function of several variables, limit of function of several variables, iterated limits, limit and path, continuity of function of several variables; Differentiability of Function of Several Variables: directional derivatives, Introduction to partial derivatives, notations and geometric interpretation, higher order partial derivatives and problems, differentiability of function of two variables, theorems on differentiability conditions, chain rules for differentiability, derivatives of implicit functions, homogeneous functions, Euler’s theorem for homogeneous functions of n-variables, Extreme values of functions of two variables, Lagrange’s method of undetermined multipliers; Multiple Integrals: Double integration over rectangular region and nonrectangular region, double integrals in polar co-ordinates, triple integrals, Volume by triple integrals, cylindrical and spherical co-ordinates, change of variables in double integrals and triple integrals. References:

1. S. Narayan & P.K. Mittal, Differential Calculus, S. Chand Publication, New Delhi, 2011. 2. T. M. Apostol, Advanced Calculus Volume II, Wiley India Publication, Delhi, 2007. 3. S. R. Ghorpade & B. V. Limaye, A course in Multivariable Calculus & Analysis, Springer India, 2014. 4. David V. Widder, Calculus, PHI, New Delhi, India, 2012. 5. M. J. Strauss, G. L. Bradley and K. J. Smith, Calculus, Dorling Kindersley Pvt.Ltd., New Delhi, 2007.

MA2212: PARTIAL DIFFERENTIAL EQUATIONS & SYSTEM OF ODE [3 1 0 4] Partial Differential Equations: Formation, order & degree, linear and non-linear partial differential equations of the first order, complete solution, singular solution, general solution, solution of Lagrange’s linear equations, charpit’s general

method of solution; Linear Partial Differential Equations of Second and Higher Orders: Linear and non-linear homogeneous and non-homogeneous equations with constant coefficients, partial differential equation with variable coefficients reducible to equations with constant coefficients, their complimentary functions and particular integrals, equations reducible to linear equations with constant coefficients. classification of linear partial differential equations of second order, hyperbolic, parabolic and elliptic types, reduction of second order linear partial differential equations to canonical (normal) forms and their solutions, solution of linear hyperbolic equations, Monge’s method for partial differential equations of second order. Cauchy’s problem for second order partial differential equations, characteristic equations and characteristic curves of second order partial differential equation, method of separation of variables; Systems of Linear Differential Equations: Types of linear systems, differential operators, an operator method for linear systems with constant coefficients, basic theory of linear systems in normal form, homogeneous linear systems with constant coefficients, two equations in two unknown functions, the method of successive approximations. References:

1. D. A. Murray, Introductory Course on Differential Equations, Orient Longman, 2005. 2. M. D. Raisinghania, Ordinary and Partial differential equations, S. Chand, India, 2018. 3. Frank Ayres, Theory and Problems of Differential Equations, McGraw Hill Book Company, 1972. 4. A.R. Forsyth, A Treatise on Differential Equations, Macmillan and Co. Ltd. 5. I. N. Sneddon, Elements of Partial Differential Equations, Tata McGraw Hill Book Company, 1998. 6. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York, 2016. 7. S. L. Ross, Differential equations, 3rd edition, John Wiley and Sons, India,2004

MA2213: LINEAR ALGEBRA [3 1 0 4] Vector spaces: Subspaces, linear dependence, independence, linear span and basis, dimension of a vector space;

Linear Transformations: definition, some results on linear operator, different types of transformations, rank and nullity,

singular and non-singular transformations, inverse linear transformation, isomorphism between vector spaces, linear

mapping, composition of linear maps; Matrices: Symmetric, skew symmetric matrices, hermitian and skew hermitian

matrices, row and column matrices, elementary operations on matrices, rank of a matrix; eigen values, eigen vectors

and the characteristic equation of a matrix, Cayley Hamilton theorem and its application in finding inverse of a matrix,

applications of matrices to a system of linear equations (both homogeneous and non-homogeneous), theorems on

consistency of a system of linear equations; Representation of Transformations by Matrices: Introduction,

determination of linear transformation for a given matrix and bases, matrix identity and zero transformations, linear

operations on Mmn, matrix of the composition of linear transformations, polynomials of a linear transformation, rank

and nullity of matrix, range of a matrix, kernel of a matrix, matrix of change of basis.

References:

1. K. B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt. Ltd, New Delhi, 2007.

2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, First course in Linear Algebra, New Age International Ltd,

2012.

3. K. Hoffman and R. Kunze, Linear Algebra, 2nd edition, Prentice Hall, Englewood Cliffs, New Jersey, 2014.

4. S. Kumaresan, Linear Algebra-A geometric approach, Prentice Hall of India, 2000.

5. R. B. Dash and D. K. Dalai, Fundamentals of Linear Algebra, Himalaya Publishing house, 2008.

6. Serge Lang, Linear Algebra, 3rd edition, Springer-Verlag, New York 2005.

MA2214: VECTOR CALCULUS AND STATICS [3 1 0 4]

Vector Algebra: Addition, scalar multiplication, scalar products, vector product, scalar and vector triple products, product of four vectors, reciprocal vectors, geometrical applications, vector equations of lines and planes, parametric representation of a curve, the circle and other conic sections, notions of a vector function of a single variable; Vector Calculus: Vector differentiation, total differential, gradient, divergence and curl, directional derivatives, Laplacian operator; Vector Integration: Path, line, surface and volume integrals, line integrals of linear differential forms, integration of total differentials, conservative fields, conditions for line integrals to depend only on the end-points, fundamental theorem on exact differentials, theorems of Green, Gauss, Stokes, and problems based on these; Statics: Forces, couples, co-planar forces, static equilibrium, friction, equilibrium of a particle on a rough curve, virtual work; catenary, forces in three dimensions, reduction of a system of forces in space, invariance of the system, general conditions of equilibrium, center of gravity for different bodies, stable and unstable equilibrium. References:

1. A.R. Vasishtha, Text Book on Vectors, Krishna Prakashan, Meerut, U.P., India, 2014. 2. S. Narayan and P. K. Mittal, A Text Book of Vector Calculus, S. Chand & Company Pvt. Ltd, New Delhi, 2009. 3. A. S. Ramsey, Statics, Cambridge University Press, 2nd edition, 2009. 4. S. L. Loney, The Elements of Statics and Dynamics Part-I, Aitbs publication, 2016. 5. J. E. Marsden and A. Tromba, Vector Calculus, 5th edition, W. H. Freeman, 2003. 6. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, Wiley India Pvt. Ltd., 2010. 7. T. Apostal, Calculus, Vol. I&II, 2nd Edition, Wiley Students Edition, India, 2012.

MA2215: MATHEMATICAL MODELING [3 1 0 4]

Power Series: solution of a differential equation about an ordinary point, solution about a regular singular point, Bessel’s equation and Legendre’s equation; Laplace Transform and Inverse transform: application to initial value problem up to second order; Monte Carlo Simulation Modeling: simulating deterministic behavior (area under a curve, volume under a surface), Generating Random Numbers: middle square method, linear congruence; Queuing Models: harbor system, morning rush hour, overview of optimization modeling; Linear Programming Model: Geometric solution algebraic solution, simplex method, sensitivity analysis. References:

1. T. Myint-U and L. Debnath, Linear Partial Differential Equation for Scientists and Engineers, Springer, Indian reprint, 2008.

2. M. M. Meerschaert, Mathematical Modeling, Academic Press, 4th edition, 2013. 3. F. R. Giordano, M. D. Weir and W. P. Fox, A First Course in Mathematical Modeling, Thomson Learning,

London and New York, 2010. 4. T. Witelski, Methods of Mathematical Modelling: Continuous Systems and Differential Equations, 1st edition,

Springer, 2015.

MA3101: NUMERICAL ANALYSIS [2 1 0 3] Errors in Numerical Computing: Introductions, types of errors; Numerical Solution of Algebraic and Transcendental Equations: Bisection method, Regula falsi method, Secant method, Iteration method, Newton-Raphson method, Bairstow’s method, synthetic division scheme; Solution of Linear System of Equations: Gauss elimination method, Gauss Jordan method, Crout’s method, Cholesky method, Gauss Jacobi method, Gauss Seidel method; Finite Differences: Finite difference operators, relation between difference operators, Lagrange and Newton-divided interpolation, Newton -Gregory forward and backward interpolation, central interpolation, Stirling formulae; Numerical Differentiation and Integration: Numerical differentiation of Newton’s forward and backward formula, Newton’s Cotes-quadrature formula, numerical integration by trapezoidal rule, Simpson’s rule – 1/3rd and 3/8th rule. Weddle rule; Numerical Solution of Ordinary Differential Equations (for first order only): Picard’s method, Euler’s method, Modified Euler method, Taylor series method, Runge-Kutta methods. References:

1. G. Haribhaskaran, Numerical Methods, Laxmi Publications, Delhi, 2008. 2. M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerical Methods for Scientific and Engineering Computation,

New Age International, 2013. 3. H.C Saxena, Finite Differences and Numerical Analysis, S. Chand & Company Ltd., New Delhi, 2015. 4. M. K. Venkataraman, Numerical Methods in Science and Engineering, National Publishing Company, 6th

Edition, 2012. 5. K. Sankara Rao, Numerical Methods for Scientists and Engineers, 2nd edition, Prentice Hall India, 2004. 6. P. Kandasamy, Numerical Methods, S. Chand & Co., New Delhi, 2007.

MA3102: OPERATIONS RESEARCH [2 1 0 3] Operations Research: Origin, definition and scope; Stochastic Process: Introduction, basic concept, Poisson process, Birth-death process, Queuing Models: Basic components of a queuing system, steady-state solution of Markovian queuing models with single and multiple servers (M/M/1. M/M/C, M/M/1/k, M/MC/k ); Replacement Problems: Introduction, items that deteriorate, items that fail completely/suddenly, system reliability and failure rate; Sequencing: Introduction, solution of processing n jobs through 2 machines, n jobs through 3 machines and n jobs through m machines; Inventory Control Models: Economic order quantity(EOQ) model with uniform demand, EOQ when shortages are allowed, EOQ with uniform replenishment, inventory control with price breaks; Game Theory: Two person zero sum game, game with saddle points, rule of dominance, algebraic, graphical and linear programming methods for solving mixed strategy games. References:

1. J. K. Sharma, Operations Research and Application, Mc. Millan and Company, New Delhi, 2012. 2. S.D, Sharma, Operations Research, Kedar Nath Ram Nath & Co., 2010. 3. H. A. Taha, Operations Research -An introduction, Prentice Hall of India Pvt. Ltd. New Delhi, 2003. 4. B. Mahadevan, Operation Management: Theory and Practice, Pearson Education India, 2015. 5. K. Swarup, P.K. Gupta and M. Mohan, Operation Research, Sultan Chand & Sons, 2010.

MA3103: DYNAMICS [3 1 0 4] Kinematics and Kinetics: Fundamental notions and principles of dynamics, Laws of motion, Relative velocity; Kinematics: Radial, transverse, tangential, normal velocities and accelerations, Simple harmonic motion, repulsion from a fixed pint, motion under inverse square law, Hooke’s law, horizontal and vertical elastic strings, motion on an inclined plane, motion of a projectile, work, energy and impulse, conservation of linear momentum, principle of conservation of energy, uniform circular motion, motion on a smooth curve in a vertical plane, motion on the inside of a smooth vertical circle, cycloidal motion, motion in the resisting medium, resistance varies as velocity and square of velocity; Central Forces: Stability of nearly circular orbits, Kepler’s laws, time of describing an arc and area of any orbit, slightly disturbed orbits. References:

1. P. S. Deshwal, Particle Dynamics, New Age International, New Delhi, 2000. 2. M. Ray and G. C. Sharma, A Text Book on Dynamics, S. Chand and Co., 2010.

3. A. S. Ramsey, Dynamics, Cambridge University Press, 2009. 4. S. L. Loney, An Elementary Treatise on the Dynamics of a Particle, Cambridge University Press, 2013.

MA3130: LAB ON NUMERICAL ANALYSIS [0 0 2 1] The following practical will be performed using C language: Bisection method, Regula falsi method, Secant method, Iteration method, Newton-Raphson method, Gauss elimination method, Gauss Jordan method, Crout’s method, Cholesky method, Gauss Jacobi method, Gauss Seidel method, Lagrange and Newton interpolation, Gregory-Newton forward and backward difference interpolation, Central interpolation, Stirling formula, trapezoidal rule, Simpson’srule – 1/3rd and 3/8th rule, Weddle rule, Picard’s method, Euler’s method, Modified Euler method, Taylor series method, Runge-Kutta methods. Reference:

1. K. Das, Numerical Methods Theoretical and Practice, U.N. Dhur & Sons, 2nd edition, 2011.

MA3131: LAB ON OPERATIONS RESEARCH [0 0 2 1] The following practical will be performed using software: Game theory, Network Analysis-PERT and CPM, Sequencing Problems, Queuing models. References:

1. M.W. Carter and Camille C, Operation Research: A Practical Introduction, CRC Press, 1st edition, 2000. MA3201: COMPLEX ANALYSIS [3 1 0 4] Analytic Functions: (Recapitulation) functions of complex variables, mappings, limits, continuity, derivatives, C-R equations, analytic functions; Complex Integration: Complex valued functions, contour, contour integrals, Cauchy- Goursat theorem, Cauchy integral formula, Moreras theorem, Liouvilles theorem, fundamental theorem of algebra; Power Series: Convergence of sequences and series, power series and analytic functions, Taylor series, Laurent’s series, absolute and uniform convergence, integration and differentiation of power series, uniqueness of series representation, zeros of an analytic function classification of singularities, behavior of analytic function at an essential singular point; Residues and Poles: Residues, Cauchy – Residue theorem, residues at poles, evaluation of improper integrals, evaluation of definite integrals, the argument principle, Rouche’s theorem, Schwarz lemma, maximum modules principle, minimum modules principle, complex form of equations of straight lines, half planes, circles, etc., analytic (holomorphic) function as mappings; conformal maps; Transformations: Mobius transformation, cross ratio, symmetry and orientation principle, examples of images of regions under elementary analytic function. References:

1. R.V. Churchill, J.W. Brown, Complex Variables and Applications, 8th edition, McGraw Hill Series, 2008. 2. B. Choudary, The element of Complex Analysis, 2nd edition, Wileys Eastern Ltd., 1983. 3. J. B. Conway, Functions of one complex variable, Springer International Student edition, Narosa Publishing

House, 2000. 4. A. R. Shastri, An Introduction to Complex Analysis, Macmillan India Ltd., 2003. 5. W. Rudin, Real and Complex Analysis, McGraw Hill Series, 1987.

MA3202: RIEMANN INTEGRATION & SERIES OF FUNCTIONS [3 1 0 4] Riemann Integration: Inequalities of upper and lower sums, Riemann conditions of integrability, Riemann sum and definition of Riemann integral through Riemann sums, equivalence of two definitions, Riemann integrability of monotone and continuous functions, properties of the Riemann integral, definition and integrability of piecewise continuous and monotone functions, intermediate value theorem for Integrals, fundamental theorems of Calculus; Improper Integrals: Convergence for finite and infinite limits, Convergence of Beta and Gamma functions, pointwise and uniform convergence of sequence of functions; Theorems on Continuity: derivability and integrability of the limit function of a sequence of functions, Series of Functions: Theorems on the continuity and derivability of the sum function of a series of functions, Cauchy criterion for uniform convergence and Weierstrass M-Test, limit superior and limit inferior; Power series: radius of convergence, Cauchy Hadamard theorem, differentiation and integration of power series. References:

1. G. Das and S. Pattanayak, Fundamentals of mathematics analysis, TMH Publishing Co., 2016 2. S.C. Mallik and S. Arora, Mathematical analysis, New Age International Ltd., New Delhi, 2012. 3. K. A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer,

Indian reprint, 2004. 4. S. Narayan, Elements of real analysis, S. Chand & Co. 2017. 5. R. G. Bartle D.R. Sherbert, Introduction to Real Analysis, 3rd edition, John Wiley and Sons Asia, Pvt. Ltd.,

Singapore, 2002. 6. C. G. Denlinger, Elements of Real Analysis, Jones & Bartlett, 2011. 7. W. Rudin, Real and Complex Analysis, McGraw Hill Series, 1987.

MA3203: METRIC SPACE [3 1 0 4] Basic Definition: metric spaces, open spheres and closed spheres, neighbourhood of a point, open sets, interior points, limit points, closed sets and closure of a set, boundary points, diameter of a set, subspace of a metric space, convergent

and Cauchy sequences, complete metric space, dense subsets and separable spaces, nowhere dense sets, continuous functions and their characterizations; Isometry and Homeomorphism: Compact spaces, sequential compactness and Bolzano-Weierstrass property, finite intersection property, continuous functions and compact sets. disconnected and connected sets, components, continuous functions and connected sets. References:

1. S. Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag London, 2009. 2. B. K. Tyagi, First Course in Metric Spaces, Cambridge University Press, 2010. 3. K.C. Sarangi, Real Analysis and Matric Spaces, Ramesh Book Depot, 2016. 4. P.K. Jain and Khalil Ahmad, Metric spaces, Second Edition, Narosa Publishing House, New Delhi, 2003. 5. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw Hill, 1963. 6. E.T. Copson, Metric spaces, Cambridge University Press, 1968. 7. S. Kumaresan, Topology of Metric Spaces, 2nd edition, Narosa Publishing House, 2011.

DSE – I & LAB MA2240: STATISTICAL INFERENCE [2 1 0 3] Estimation: Parametric space, sample space; Point Estimation: Properties of good estimator: Consistency, unbiasedness, efficiency, sufficiency. Neymann factorization theorem, complete sufficient statistics, minimum – variance unbiased (MVU) estimators, exponential family of distributions and its properties, Cramer- Rao inequality, minimum variance bound (MVB) estimators; Interval Estimation: Confidence intervals for the parameters of various distributions, confidence intervals for difference of means and for ratio of variances; Methods of Estimation: Method of maximum likelihood, methods of moments; Elements of Statistical Decision Theory: Neyman theory of testing of hypotheses, simple and composite hypotheses, null and alternative hypotheses, two types of errors, critical region, level of significance, power of the test, unbiased tests, Neyman- Pearson lemma, construction of most powerful test, uniformly most powerful test, uniformly most powerful unbiased test; Tests of Significance: tests of significance based on t, F and Chi-square distributions. References:

1. A.M. Goon, M.K. Gupta and B. Dasgupta, An Outline of Statistical Theory, Vol. II, 3rd edition, World Press, Kolkata, 2005.

2. M Kendall, A. Stuart and J.K. Ord, Kendall's Advanced Theory of Statistics, Oxford University Press, 5th edition, 1991.

3. P. Mukhopadhyay, Applied Statistics, Books & Allied Ltd., 2011. 4. G. Casella, and R.L. Berger, Statistical Inference, Second Edn. Thomson Duxbury, 2002. 5. R.V. Hogg, and E.A. Tanis, Probability and Statistical Inference, 9th edition, Macmillan Publishing Co. Inc.,

2014. 6. V. K. Rohatgi, Statistical Inference, John Wiley and Sons, 2003.

MA2230: LAB ON STATISTICAL INFERENCE [0 0 2 1] The following practical will be performed using statistical software: Method of maximum likelihood, methods of moments, minimum chi- square and modified minimum chi- square, computation of confidence intervals for the parameters of various distributions, confidence intervals for difference of means and for ratio of variances, confidence interval for binomial proportion and population correlation coefficient when population is normal. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. Gopal K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

MA2241: INVENTORY THEORY AND DYNAMIC PROGRAMMING [3 1 0 4] Inventory Control: Different variables involved. Single item deterministic- economic lot size models with uniform rate, finite & infinite production rates, with or without shortage-multiitem models with one constant; Deterministic Models with Price-Breaks: aii units discount model and incremental discount model. Probabilistic single period profit maximization models with uniform demand, instantaneous demand, with or without setup cost, dynamic inventory models, multi-echelon problems. Integrated approach to production inventory and to maintenance problems, feedback control in inventory management; Dynamic Programming: Bellman's principle of optimality, characteristics of a dynamic programming problem, solutions of simple classical problems with single constraint. Solution to linear programming problem and integer programming problem using dynamic programming approach; Applications of Dynamic Programming: Shortest path through a network, production planning, inventory problems, investment planning, cargo loading and knapsack problems. References:

1. J. K. Sharma, Operations Research: Theory and Applications, McMillan India Ltd. 2013. 2. S. Axsater. Inventory Control, International Series in Operations Research & Management Science. Springer

pub., 2nd edition. 2006. 3. Zipkin, Foundations of Inventory Management, Mc-Graw Hill Inc., 2000. 4. R. E. Larson and J. l. casti, "Principles of Dynamic Programming", 1st edition, 1982. 5. F. Naddor, Inventory System, John Wiley & Sons, Inc. New York, 1966.

MA2242: INFORMATION THEORY [3 1 0 4] Introduction: Mathematical theory of information theory in communication system, entropy and its properties, measures of information, self-information, mutual information, average information, Csiszar’s f-divergence measure and their applications; Discrete Memoryless Channels: Classification of channels, calculation of channel capacity, decoding scheme-the ideal observer, the time-discrete Guassian channel, fundamental theorem of information theory, uncertainty of absolutely continuous random variable, the converse to the coding theorem for time-discrete Gaussian channel, the time-continuous Gaussian channel, band–limit channels; Source Model and Coding: Channels model and coding, unique decipherable codes and problems, condition of instantaneous codes, code word length, kraft inequality, noiseless coding theorem, Construction of Codes: Shannon Fano, Shannon binary and Huffman codes; Error Correcting Codes: Minimum distance principle, relation between distance and error correcting properties of codes, the hamming bound, parity check coding, bounds on the error correcting ability of parity check codes. References:

1. R.G. Gallager, Information theory and Reliable Communication, Wiley India, 2002. 2. R. Ash, Information Theory, John Wiley & Sons, 2012. 3. W.W. Peterson and E.J. Weldon, Error Correcting Codes, MIT Press, 1972. 4. R. Bose, Information Theory, Coding and Crptography, TMH, 2007. 5. S. Gravano, Introduction to Error Control Codes, Oxford University Press, 2007.

MA2243: MATHEMATICAL FINANCE [3 1 0 4] Basic Principles: Comparison, arbitrage and risk aversion, interest (simple and compound, discrete and continuous), time value of money, inflation, net present value, internal rate of return (calculation by bisection and Newton-Raphson methods), comparison of NPV and IRR, bonds, bond prices and yields, Macaulay and modified duration; Term Structure of Interest Rates: spot and forward rates, explanations of term structure, running present value, floating-rate bonds, immunization, convexity, putable and callable bonds, asset return, short selling, portfolio return, (brief introduction to expectation, variance, covariance and correlation), random returns, portfolio mean return and variance, diversification, portfolio diagram, feasible set; Markowitz Model: (review of Lagrange multipliers for 1 and 2 constraints), Two fund theorem, risk free assets, one fund theorem, capital market line, Sharpe index, Capital Asset Pricing Model (CAPM), betas of stocks and portfolios; Security Market Line: use of CAPM in investment analysis and as a pricing formula, Jensen’s index. References:

1. D. G. Luenberger, Investment Science, Oxford University Press, Delhi, 1998.

2. J. C. Hull, Options, Futures and Other Derivatives, 6th Ed., Prentice-Hall India, Indian reprint, 2006.

3. S. Ross, An Elementary Introduction to Mathematical Finance, 2nd Ed., Cambridge University Press, USA,

2003.

DSE – III (A) MA3140: BASIC ECONOMETRICS [2 1 0 3] Introduction: Definition and scope of econometrics, methodology of econometric research; Simple Linear Regression Model: Assumptions, estimation (through OLS method), desirable properties of estimators, Gauss- Markov theorem, interpretation of regression coefficients, testing of regression coefficients, test for regression as a whole, coefficient of determination; Multiple Regression Analysis: problem of estimation and problem of inference; Problems in OLS Estimation: Multicollinearity, nature, consequences, detection and remedial measures. Heterosedasticity: Nature, consequences, detection and remedial measures, autocorrelation: concept, consequences of autocorrelated disturbances, detection of autocorrelation, their estimation and testing, estimation using Durbin-Watson statistic. References:

1. S.P. Singh, A.K Parashar and H.P. Singh, Econometrics, S. Chand and Company Ltd, New Delhi, 2000. 2. D.N. Gujarati, Basic Econometrics, 4th edition, McGraw−Hill, New Delhi, 2004. 3. W. H. Greene, Econometric Analysis, Pearson, 8th edition, 2017. 4. W.F. Griffith, R.H. Hill and G.G. Judge, Learning and Practicing Econometrics, John Wiley, New York, 1993. 5. J. Johnston, Econometric Methods, McGraw Hill, New York, 4th edition, 1997. 6. G.S. Maddala, and K Lahiri, Introduction to Econometrics, Wiley, 2009.

MA3141: MODELLING & SIMULATION [2 1 0 3] Introduction to Modelling and Simulation: System analysis, classification of systems, system theory basics, its relation to simulation, model classification, conceptual, abstract, and simulation models. heterogeneous models, methodology of model building, simulation systems and languages, means for model and experiment description, Principles of simulation system design, parallel process modelling, using petri nets and finite automata in simulation models of queuing systems, discrete simulation models, model time, simulation experiment control; Continuous System Modelling: Overview of numerical methods used for continuous simulation, combined simulation, role of simulation in digital systems design, special model classes, models of heterogeneous systems; Checking Model Validity: verification of models, analysis of simulation results, simulation results visualization, interactive simulation, design and control of simulation experiments; Model Optimization: Generating, transformation, and testing of pseudorandom numbers, stochastic models, Monte Carlo method, overview of commonly used simulation systems. References:

1. S. Ross, Simulation, Academic Press, 5th edition, 2012. 2. A. Law and D. Kelton, Simulation Modelling and Analysis, McGraw-Hill, 5th edition, 2014.

3. P. Fishwick, Simulation Model Design and Execution, Prentice Hall, 1995. 4. B. P. Zeigler, Theory of modelling and simulation, 3rd edition, 2018.

DSE – III (B) MA3142: RELIABILITY MODELING AND ANALYSIS [2 1 0 3] Reliability Basic Concepts: Concept of reliability, early age failures, wear-out failures and chance failures, derivation of general reliability function failure rate, failure density function and mean time between failures; System Reliability Evaluation: series system, parallel system, partially redundant system, standby system with perfect switching / imperfect switching, effect of spare components (identical / non- identical) on the system reliability, Wear-out and Component reliability, combined effect of wear-out and chance failures, reliability of a two component system with single repair facility; Reliability Evaluation Techniques: Conditional probability approach, cut set method, approximation evaluation, deducing the minimal cut sets, tie set method, connection matrix technique. References:

1. J. Medli, Stochastic Processes, New Age International Publisher, 1996. 2. E. Balagurusamy, Reliability Engineering, Tata McGraw-Hill, 2010. 3. S. Zack, Introduction to Reliability Analysis: Probability Model and Statistical Methods, Springer Verlag, 1992. 4. B.K. Kale and S.K. Sinha, Life Testing and Reliability Estimation, 1980.

MA3143: PORTFOLIO OPTIMIZATION [2 1 0 3] Financial Markets: Investment objectives, measures of return and risk, types of risks, risk free assets, mutual funds, portfolio of assets, expected risk and return of portfolio; Diversification: Mean-variance portfolio optimization- the Markowitz model and the two-fund theorem, risk-free assets and one fund theorem, efficient frontier, portfolios with short sales; Capital Market Theory: Capital assets pricing model- the capital market line, beta of an asset, beta of a portfolio, security market line; Index Tracking Optimization Models: Portfolio performance evaluation measures. References:

1. F. K. Reilly, Keith C. Brown, Investment Analysis and Portfolio Management, 10th edition, South-Western Publishers, 2011.

2. H.M. Markowitz, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Blackwell, New York, 1987. 3. M.J. Best, Portfolio Optimization, Chapman and Hall, CRC Press, 2010. 4. D.G. Luenberger, Investment Science, 2nd edition, Oxford University Press, 2013.

DSE – IV (A)

MA3241: MULTIVARIATE ANALYSIS [2 1 0 3] Multivariate Normal Distribution: Introduction to multivariate statistical methods, matrix algebra, multiple regression formulated in matrix terms, multivariate normal distribution, maximum likelihood estimators of mean vector and covariance Matrix; Wishart Distribution: its matrix and properties, Hotelling's T2 statistic, derivation and its distribution uses of T2 statistic, Beheran -Fisher’s problem. Multivariate Linear Regression Model: Estimation of parameters and their properties, distribution of the matrix of sample regression coefficients, test of linear hypothesis about regression coefficients, multivariate analysis of variance (MANOVA) of one way classified data. Wilk’s lambda criterion, likelihood ratio test criteria for testing independence of sets of variables, likelihood ratio criteria for testing equality of covariance matrices and identity of several multivariate normal populations, Fisher’s discriminant function, Mahalanobis’ distance; Principle Component Analysis: Principal components, its uses and importance, canonical variables and canonical correlations. References:

1. T.W. Anderson, An Introduction to Multivariate Statistical Analysis, John Wiley, 2016. 2. C. R. Rao, Linear Statistical Inference and its Applications, John Wiley, 2015. 3. R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, Prentice Hall of India, 2001. 4. K. V. S. Sarma and R. V. Vardhan. Multivariate Statistics Made Simple: A Practical Approach, CRC Press,

2018. 5. A. C. Rencher, Methods of Multivariate Analysis, 2nd edition, John Wiley & Sons, 2002. 6. B. G. Tabachnick and L. S. Fidell, Using Multivariate Statistics, 5th edition, Boston, MA: Allyn & Bacon, 2007.

MA3242: NUMERICAL METHODS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS [2 1 0 3] Ordinary Differential Equations: Initial value problems and existence theorem, truncation error, deriving finite difference equations, single step methods for first order initial value problems, Taylor series method, Euler method, Picard’s method of successive approximation, Runge Kutta methods, stability of single step methods, multi-step methods for first order initial value problem, Predictor-Corrector method, Milne and Adams Moulton Predictor corrector method, system of first order ordinary differential equations, higher order initial value problems, stability of multi-step methods, root condition; Boundary Value Problems: finite difference methods, shooting methods, stability, error and convergence analysis, nonlinear boundary value problems; Partial Differential Equations: Classification, Finite difference approximations to partial derivatives, solution of one dimensional heat conduction equation by explicit and implicit schemes, stability and convergence criteria, Laplace equation using standard five point formula and diagonal five point formula, Iterative methods for solving the linear systems, hyperbolic equation, explicit and implicit schemes, method of characteristics, solution of wave equation, solution of first order hyperbolic equation, Von Neumann stability.

References:

1. K. E. Atkinson, W. Han and D. E. Stewart, Numerical Solution for Ordinary Differential Equations, John Wiley & Sons, New York, 2011.

2. M K Jain, S R K Iyengar and R K Jain, Numerical Methods for Scientific and Engineering Computation, New Age International Publication, New Delhi, 2014.

3. G. D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, London, 1986.

DSE – IV (B) MA3243: MECHANICS [2 1 0 3] Moment: Moment of a force about a point and an axis, couple and couple moment, Moment of a couple about a line, resultant of a force system, distributed force system, free body diagram, free body involving interior sections, general equations of equilibrium, two point equivalent loading, problems arising from structures, static indeterminacy; Laws of Coulomb Friction: application to simple and complex surface contact friction problems, transmission of power through belts, screw jack, wedge, first moment of an area and the centroid, other centers, Theorem of Pappus-Guldinus, second moments and the product of area of a plane area, transfer theorems, relation between second moments and products of area, polar moment of area, principal axes; Conservative Force Field: conservation for mechanical energy, work energy equation, kinetic energy and work kinetic energy expression based on center of mass, moment of momentum equation for a single particle and a system of particles, translation and rotation of rigid bodies, Chasles’ theorem, general relationship between time derivatives of a vector for different references, relationship between velocities of a particle for different references, acceleration of particle for different references. References:

1. I.H. Shames and G. Krishna Mohan Rao, Engineering Mechanics: Statics and Dynamics, 4th edition, Pearson

Education, Delhi, 2009.

2. R.C. Hibbeler and Ashok Gupta, Engineering Mechanics: Statics and Dynamics, 11th edition, Pearson

Education, Delhi, 2011.

MA3244: INTRODUCTION TO GRAPH THEORY [2 1 0 3] Preliminaries: Graphs, isomorphism, subgraphs, matrix representations, degree, operations on graphs, degree sequences; Connected Graph and Shortest Path: Walks, trails, paths, connected graphs, distance, cut-vertices, cut-edges, blocks, connectivity, weighted graphs, shortest path algorithms; Trees: Characterizations, number of trees, minimum spanning trees; Special Classes of Graph: Bipartite graphs, line graphs, chordal graphs; Eulerian Graph: Characterization, Fleury’s algorithm, chinese-postman-problem; Hamilton Graphs: Necessary conditions and sufficient conditions independent sets, coverings; Matching: Basic equations, matchings in bipartite graphs, perfect matchings, greedy and approximation algorithms; Vertex Coloring: Chromatic number and cliques, greedy coloring algorithm, coloring of chordal graphs; Planar Graphs: Basic concepts, Eulers formula; Directed Graph: Out-degree, in-degree, connectivity, orientation, Eulerian directed graphs, Hamilton directed graphs. References:

1. D. B. West, Introduction to Graph Theory, Prentice Hall of India, 2012. 2. N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, 2009. 3. G. Chatrand and Ping Zhang, Introduction to Graph Theory, McGraw Hill Education, 2017. 4. R. J Wilson, Graph Theory, Prentice Hall, 2010.

GENERIC ELECTIVES

GE – I & LAB

MA1140: DESCRIPTIVE STATISTICS [2 1 0 3] Introduction of Statistics: Definition, scope, uses and limitations; Types of Data: Qualitative and quantitative data, nominal and ordinal data, time series data, discrete and continuous data, frequency and non-frequency data; Collection of Data: Collection of primary and secondary data- its major sources, classification and tabulation of data; Presentation of Data: Frequency distribution and cumulative frequency distribution, diagrammatic and graphical presentation of data, construction of bar, pie diagram, histogram, frequency polygon, frequency curve and ogives; Measures of Central Tendency and Location: Arithmetic mean, median, mode, geometric mean, harmonic mean, partition values-quartiles, deciles, percentiles and their graphical location along with their properties, applications, merits and demerits; Measures of Dispersion: Characteristics for an ideal measure of dispersion, absolute and relative measures of dispersion, range, inter quartile range, quartile deviation, coefficient of quartile deviation, mean deviation, coefficient of mean deviation, standard deviation, coefficient of variation and properties of these measures; Moments, Skewness and Kurtosis: Moments about mean and about any point and their relationship, effect of change of origin and scale, Sheppard’s correction for moments (without derivation), Charlier’s checks, coefficients of skewness and kurtosis with their interpretations; Bivariate Data: Scatter diagram, correlation, product moment correlation coefficient and their uses, rank correlation, concept of multiple correlation and partial correlation in case of three variables, regression analysis. References:

1. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand and Co., 3rd edition, New

Delhi, 2008.

2. V.K. Rohtagi and A K.MD.E Saleh, An Introduction to Probability & Statistics, John Wiley & Sons, 2011.

3. A.M. Goon, M.K. Gupta, B. Dasgupta, Fundamental of Statistics, Vol.-I, World Press, 2016.

4. A.M. Mood, F.A. Greybill, and D. C. Bose, Introduction to the Theory of Statistics, McGraw Hill, 2001.

MA1130: LAB ON DESCRIPTIVE STATISTICS [0 0 2 1] The following practical will be performed using statistical software: Graphical and diagrammatic representation of data, finding the measure of central tendency and dispersion for various types of data, computation of moments, skewness and kurtosis of data, computation of Karl Pearson’s, partial and multiple correlation coefficient and Spearman’s rank correlation coefficient, method of least squares, fitting of straight line, parabola and exponential curves, fitting of lines of regression, fitting of regression plane for three variates. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. Gopal K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY1160: GENERAL CHEMISTRY-I [2 1 0 3] Structure and Bonding: Hybridization, interactions, resonance, aromaticity, H-bonds. Mechanism: Notations, bond cleavage, electrophiles and nucleophiles, intermediates, free radicals. Stereochemistry: Isomerism, symmetry, chirality, projections, D&L- E&Z- R&S- nomenclature. Basic Concepts of Inorganic Chemistry: Structure, periodicity, ionic solids. Bonding: Covalent bonds, hybridization, VSEPRT, VBT, MOT. s-block Elements: Comparison, diagonal relationships, hybrids. Miscellaneous: Oxidation and reduction, acids and bases, noble gasses, radioactivity. References:

1. J. D. Lee, Concise Inorganic Chemistry, Blackwell Science, 2008. 2. J. E. Huheey, E. A. Keiter & R. L. Keiter, Inorganic Chemistry: Principles of Structure and Reactivity, Pearson

India, 2008. 3. G. W. Solomon and B. F. Craig, Organic Chemistry, John Wiley & Sons, Inc., 2010.

4. P. Sykes, A Guidebook to Mechanism in Organic Chemistry, Pearson India, 2003.

CY1138: ORGANIC CHEMISTRY LABORATORY [0 0 2 1] Basics: Distillation, crystallization, decolourization and crystallization using charcoal, sublimation. Qualitative Analysis: Identification, functional group analysis, melting point, preparation of derivatives. Reference:

1. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book Agency, 2011.

PY1160: MECHANICS AND SPECIAL THEORY OF RELATIVITY [2 1 0 3] Dynamics of a System of Particles: Centre of mass, conservation of momentum, idea of conservation of momentum from Newton’s third law, impulse, motion of rocket, potential energy, stable and unstable equilibrium, elastic potential energy, work-energy theorem, work done by non-conservative forces, law of conservation of energy, elastic and inelastic collisions between particles. Rotational Dynamics: Angular momentum of a particle and system of particles, conservation of angular momentum, rotation about a fixed axis, moment of inertia, kinetic energy of rotation, motion involving both translation and rotation. Gravitation and Central Force Motion: Law of gravitation, inertial and gravitational mass, gravitational potential energy, potential and field due to spherical shell and solid sphere. motion of a particle under central force field, two body problem and its reduction to one body problem and its solution, the energy equation and energy diagram, orbits of artificial satellites. Inertial and Non-inertial systems: Reference frames, inertial frames and Galilean transformations non-inertial frames and fictitious forces, uniformly rotating frame, physics laws in rotating coordinate systems, centrifugal forces, Coriolis force, components of velocity and acceleration in cylindrical and spherical coordinate systems. Special Theory of Relativity: Postulates, Michelson-Morley experiment, Lorentz transformations, simultaneity and order of events, Lorentz contraction, variation of mass with velocity, rest mass, massless particles, mass-energy equivalence, relativistic Doppler effect, transformation of energy and momentum. References:

1. D. Kleppner, R. J. Kolenkow, An introduction to mechanics, Tata McGraw-Hill, 2007. 2. D. S. Mathur, Mechanics, S. Chand & Company Limited, 2014. 3. M. R. Spiegel, Theoretical Mechanics, Tata McGraw-Hill, 2017. 4. C. Kittel, W. Knight, M. Ruderman, C. Helmholz, B. Moyer, Mechanics, Berkeley Physics course, Vol.-I, Tata

McGraw-Hill, 2010. 5. F. W. Sears, M. W. Zemansky, H. D. Young, University Physics, Narosa Pub. House, 2013. 6. M. Alonso, E. Finn, Physics Addison-Wesley, 2000.

PY1136: GENERAL PHYSICS LAB-I [0 0 2 1] Use of multimeter for measuring: (a) resistances, (b) a/c and dc voltages, (c) ac and dc currents, (d) capacitances, and

(e) frequencies; Test a diode and transistor using: (a) a multimeter and (b) a CRO, to measure (a) voltage, (b) frequency

and (c) phase difference using a CRO, to determine the moment of inertia of a flywheel, to determine the coefficient of

viscosity of water by capillary flow method, to determine the young's modulus of a wire by optical lever method, to

determine the elastic constants of a wire by Searle’s method, to determine “g” by bar pendulum.

References:

1. D. Chattopadhyay, P. C. Rakshit, An Advanced Course in Practical Physics, New Central Book Agency (P)

Ltd., 2012.

2. C. L Arora, BSc Practical Physics, S. Chand Publication, 2012.

3. R. K. Shukla, A. Srivastava, Practical Physics, New Age Publisher, 2006.

4. D. P. Khandelwal, A Laboratory Manual of Physics for Undergraduate Classes, Vani Publication House, New Delhi, 2000.

5. G. Sanon, B. Sc. Practical Physics, S. Chand, 2010. 6. B. L. Worsnop, H. T. Flint, Advanced Practical Physics, Asia Publishing House, 2002.

GE – II (A) & LAB

MA1240: PROBABILITY AND RANDOM VARIABLES [2 1 0 3] Probability Theory: Random experiments, sample space, event, algebra of events, Definitions of Probability, theorems on probability, Boole’s inequality, conditional probability, independent events, Bayes theorem and its applications; Random Variable: Random variable, distribution function, discrete random variable, probability mass function, distribution function of discrete random variable, continuous random variable, probability density function, distribution function of continuous random variable. joint probability mass function, marginal probability function, conditional probability function, joint distribution function, marginal distribution function Joint density function, marginal density function, stochastic independence, independent random variables; Mathematical Expectation: Definition, expected value of random variable, expected value of a function of a random variable, addition and multiplication theorems and their generalizations, covariance, expectation and variance of a linear combination of random variable, Cauchy-Schwartz inequality, conditional expectation and conditional variance; Generating Functions: Definition, limitations and properties of moment generating function, uniqueness theorem, cumulates, properties of cumulates, effect of change or origin and scale, characteristic function, properties of characteristic function, uniqueness theorem. Probability generating function. References:

1. A.M. Mood, F.A. Greybill, and D.C. Bose, Introduction to the Theory of Statistics, McGraw Hill, 2001.

2. S.C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand and Co., 3rd edition,

New Delhi, 2008.

3. P. L. Meyer, Introductory Probability and Statistical Applications, Addison-Wesley, 2017.

4. P.G. Hoel, Introduction to Mathematical Statistics, Asia Publishing House, 1984.

5. G.W. Snedecors and W.G. Cochran, Statistical Methods, Iowa State University Press, 1991.

6. A.M. Goon, M.K. Gupta and B. Dasgupta, Fundamental of Statistics, Vol. I, World Press, Calcutta, 2005.

MA1230: LAB ON PROBABILITY AND RANDOM VARIABLES [0 0 2 1] The following practical will be performed using statistical software: expected value of random variable, expected value of a function of a random variable, addition and multiplication theorems and their generalizations, covariance, expectation and variance of a linear combination of random variable, Cauchy-Schwartz inequality, conditional expectation and conditional variance, moment generating function, characteristic function and probability generating function. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. G. K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY1260: GENERAL CHEMISTRY-II [2 1 0 3] Basic Concepts: Introduction to physical chemistry. Solid, Liquid, and Gaseous State: Ideal and real gas, kinetic theory of gas, PV isotherms, velocity distribution, intermolecular forces, liquid crustals, lattice, unit cell, crystallography. Thermodynamics: Heat and work, laws of thermodynamics, enthalpy, entropy, free energy, thermochemistry. p-block Elements: Comparison, hydrides, oxides, halides, boron chemistry, carbides, fullerenes, halogens. d-block Elements: Transition metals, coordination compounds, metal-ligand bonding; References:

5. A. Bahl, S. S. Bahl, G. D. Tuli, Essentials of Physical Chemistry, S. Chand, 2016. 6. P. Atkins and J. de Paula, Atkins’s Physical Chemistry, Oxford University Press, NY, 2010. 7. J. D. Lee, Concise Inorganic Chemistry, Blackwell Science, 2008. 8. J. E. Huheey, E. A. Keiter & R. L. Keiter, Inorganic Chemistry: Principles of Structure and Reactivity, Pearson

India, 2008. CY1238: INORGANIC CHEMISTRY LABORATORY [0 0 2 1] Inorganic: Qualitative analysis of inorganic salts, volumetric analysis of inorganic mixtures, synthesis of transition metal complexes. Reference:

1. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book Agency, 2011.

PY1261: ELECTROMAGNETISM [3 1 0 4] Electric Field and Electric Potential: Electric field and lines, electric flux, Gauss’s law, Gauss’s law in differential form, calculation of E due to various charge distribution, electric potential difference and electric potential V, potential and electric field due to various charge distribution force and torque on a dipole, conductors in an electrostatic field, description of a system of charged conductors, an isolated conductor and capacitance, electrostatic energy due to various charge distribution. Electric Field in Matter: Dielectric constant, parallel plate capacitor with a dielectric, polarization charges and polarization vector, electric susceptibility, Gauss’s law in dielectrics, displacement vector D, relations between the three electric vectors, capacitors filled with dielectrics. Magnetic Effect of Currents: Magnetic Field B, magnetic force between current elements and definition of B, magnetic flux, Biot-Savart’s law: calculation of B due to various charge distribution, magnetic dipole and its dipole moment, Ampere’s Circuital law, B due to a solenoid and a toroid, curl and divergence of B, vector potential. forces on an isolated moving charge, magnetic force on a current carrying wire, torque on a current loop in a uniform magnetic field, Gauss’s law of magnetism, magnetic intensity (H), relation between B, M and H, stored magnetic energy in matter, B-H curve, Faraday’s law, Lenz’s law, self and mutual induction, single phase transformer, energy stored in a magnetic field, potential energy of a current loop. Ballistic Galvanometer: Current and charge sensitivity, electromagnetic damping, logarithmic damping, critical damping. References:

1. D. J. Griffiths, Introduction to Electrodynamics, PHI learning, 2015. 2. E. M. Purcel, Electricity and Magnetism, Tata McGraw-Hill Education, 2011. 3. J. H. Fewkes, J. Yarwood, Electricity and Magnetism, Oxford University Press, 1991. 4. D. C. Tayal, Electricity and Magnetism, Himalaya Publishing House, 2014. 5. M. Alonso, E. Finn, Physics, Addison-Wesley, 2000.

GE – II (B) & LAB

MA1241: APPLIED STATISTICS [2 1 0 3] Demand Analysis: Laws of demand and supply, price and supply elasticity of demand, partial and cross elasticity of demand, income elasticity of demand, utility function, methods of determining demand and supply curves from family budget and time series date, Leontief’s method, Pigou’s method, Engel curve and its different forms, Pareto’s law of income distribution, curves of concentration, Index Numbers: Introduction and their construction, Laspeyer’s, Paashce’s, Marshall –Edge Worth and Fisher’s index numbers, tests for index numbers, uses of index numbers, price, quantity and value relatives, link and chain relatives, chain base index numbers, cost of living index numbers; Time Series: Analysis of time series, components of time series, trend measurement by mathematical curves, polynomial, growth curves, moving average method, Spencer’s formulae, Effect of elimination of trend on other components of time series, variate difference method and its use for estimation of variance of the random component, measurement of seasonal fluctuations measurement of cyclical component, periodogram analysis; Statistical Quality Control: Control charts for variable and attributes, acceptance sampling by attributes- single, double, multiple and sequential sampling plans, concepts of average outgoing limit (AOQL), acceptable quality limit (ATI), acceptance sampling by variables-use of Dodge-Romig and other tables. References:

1. S.C. Gupta and V.K Kapoor, Fundamentals of Applied Statistics, Sultan Chand and Co., 3rd edition, New Delhi,

2008.

2. P. Mukhopadhyay, Mathematical Statistics, Books & Allied (P) Ltd., 2009.

3. D.C. Montgomery, Introduction to Statistical Quality Control, Wiley India Ltd., New Delhi, 2009.

4. A.M. Goon, M.K. Gupta and B. Das Gupta, Fundamentals of Statistics, Vol.-2, 2001.

5. P. Mukhopadhyay, Fundamental of Statistics, Vol.-2, 1999.

6. B.L. Agarwal, Basic Statistics, Wiley India Ltd., New Delhi, 2012.

MA1231: LAB ON APPLIED STATISTICS [0 0 2 1] The following practical will be performed using statistical software: Trend analysis by using method of semi-averages, method of curve fitting, method of moving average, Spencer’s 15 - point and 21 point – formulas, computation of seasonal variation indices by using ratio to trend method, ratio to moving average method, link relative method, measurement of cyclical component, periodogram analysis, estimation of parameters in ARIMA models, forecasting,

exponential and adaptive smoothing models, construct of (i) �̅� and R–chart (ii) p–chart (iii) c–chart. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. G. K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY2161: STRUCTURE OF MATERIALS [2 1 0 3] Basic Concepts: Introduction to inorganic chemistry. Structure of crystalline solids: Classification of materials, crystalline and amorphous solids crystal. Structure, symmetry and point groups, Brvais lattice, unit cells, types of close packing - hcp and ccp, packing efficiency, radius ratios; crystallographic direction and plane. Ceramics: Classification, structure, impurities in solids. Electrical Properties: Introduction, basic concept of electric conduction, free electron and band theory, classification of materials, insulator, semiconductor, intrinsic & extrinsic semi-conductors, metal, superconductor etc., novel materials. Magnetic Properties: Introduction, origin of magnetism, units, types of magnetic ordering: dia-para-ferro-ferri and antiferro-magnetism, soft and hard magnetic materials, examples of some magnetic materials with applications. Special topics: Biomaterials, nanomaterials, composite materials.

References: 1. W. D. Callister, Material Science and Engineering, An introduction, 3rd Edition, Willey India, 2009. 2. H. V. Keer, Principals of Solid State, Willey Eastorn, 2011. 3. J. C. Anderson, K. D. Leaver, J. M. Alexander, & R. D. Rawlings, Materials Science, Willey India, 2013.

CY1239: PHYSICAL CHEMISTRY LABORATORY [0 0 2 1] Physical: Determination of rate constants, conductometric titrations, thermochemistry, phase diagrams. Reference:

1. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book Agency, 2011.

PY1260: OSCILLATION AND WAVE OPTICS [2 1 0 3] Simple Harmonic Motion: Simple harmonic oscillations, oscillations having equal frequencies and oscillations having different frequencies (beats), superposition of n-collinear harmonic oscillations with equal phase differences and equal frequency differences, superposition of two mutually perpendicular simple harmonic motions with frequency ratios 1:1 and 1:2 using graphical and analytical methods. Damped Oscillations: Log decrement, forced oscillations, transient and steady states, amplitude, phase, resonance, sharpness of resonance, power dissipation and quality factor, Helmholtz Resonator. Standing Waves in a String: Fixed and free ends, analytical treatment, phase and group velocities, changes w.r.t position and time, energy of vibrating string, transfer of energy, normal modes of stretched strings. Wave Optics: Electromagnetic nature of light, definition and properties of wave front, Huygens principle, coherence. Interference: Division of amplitude and wavefront. Young’s double slit experiment. Lloyd’s Mirror, Fresnel’s biprism, interference in thin films (parallel and wedge-shaped), fringes of equal inclination and thickness, Newton’s Rings, Michelson Interferometer, Fabry-Perot interferometer. Diffraction: Fresnel diffraction, Fresnel’s half-period zones, theory of a zone plate, multiple foci, comparison of a zone plate with a convex lens, Fresnel’s integrals, Cornu’s spiral, Fresnel diffraction pattern due to a straight edge, a slit, and a wire, diffraction due to a single slit, a double sl it and a plane transmission grating, Rayleigh’s criterion, resolving power, dispersive power of grating. Polarization: Light polarization by reflection, refraction, Brewster’s Law, Malus Law, double refraction. References:

1. F. A. Jenkins, H. Elliott White, Fundamentals of Optics, Tata McGraw-Hill, 2013. 2. A. Ghatak, Optics, Tata McGraw-Hill, 2015. 3. S. Subrahmaniyam, B. Lal, M. N. Avadhanulu, A Textbook of optics, S. Chand, 2010. 4. E. Hecht, A. R. Ganesan, Optics, Pearson Education, 2002. 5. A. Al-Azzawi, Light and Optics: Principles and Practices, CRC Press, 2007. 6. M. Alonso, E. Finn, Physics Addison-Wesley, 2000.

PY1236: OPTICS LAB [0 0 2 1] Familiarization with: Schuster`s focusing; determination of angle of prism, to determine refractive index of the material of a prism using sodium source, to determine the dispersive power and Cauchy constants of the material of a prism using mercury source, to determine wavelength of sodium light using Fresnel Biprism, to determine wavelength of sodium light using Newton’s Rings, To determine the thickness of a thin paper by measuring the width of the interference fringes produced by a wedge-shaped Film, to determine wavelength of (1) Na source and (2) spectral lines of Hg source using plane diffraction grating, to determine dispersive power and resolving power of a plane diffraction grating. References:

1. D. Chattopadhyay, P. C. Rakshit, An Advanced Course in Practical Physics, New Central Book Agency (P) Ltd., 2012.

2. C. L Arora, BSc Practical Physics, S. Chand Publication, 2012. 3. R. K. Shukla, A. Srivastava, Practical Physics, New Age Publisher, 2006. 4. D. P. Khandelwal, A Laboratory Manual of Physics for Undergraduate Classes, Vani Publication House, New

Delhi, 2000. 5. G. Sanon, B. Sc. Practical Physics, S. Chand, 2010. 6. B. L. Worsnop, H. T. Flint, Advanced Practical Physics, Asia Publishing House, 2002.

GE – III (A) & LAB

MA2140: DISTRIBUTION THEORY [2 1 0 3] Discrete Probability Distributions: Bernoulli distribution, Binominal distributions, Poisson distribution, Poisson distribution as a limiting case of Binomial distribution, Negative Binominal distribution, Geometric distribution, Hyper-geometric distribution and their properties; Continuous Probability Distribution: Uniform distribution, Normal distribution, Exponential distribution, Beta distribution, Gamma distribution, Cauchy distribution and their properties; Limit Laws: Convergence in probability, almost sure convergence, convergence in mean square and convergence in distribution, weak law of large numbers (WLLN), strong law of large numbers (SLLN), De-Moivre-Laplace theorem, central limit theorem (C.L.T.) for i.i.d. variates, Liapunov theorem and applications of C.L.T. References:

1. A.M. Goon, A.K. Gupta, and B. Dasgupta, Fundamental of Statistics, Vol. I, World Press, Calcutta, 2005. 2. A.M. Mood, F.A. Greybill, and D.C. Bose, Introduction to the Theory of Statistics, McGraw Hill, 2001. 3. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand and Co., 3rd edition, New

Delhi, 2008.

4. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edition, John Wiley, 2005. 5. P. Mukhopadhyay, Mathematical Statistics, Books & Allied Ltd., 2009. 6. N.L Johnson, S. Kotz and A.W Kemp, Univariate discrete distributions, John Wiley, 1992. 7. N. L Johnson, S. Kotz and N Balakrishnan. Continuous Univariate distributions I & II, John Wiley, 1991. 8. S. Kotz, N. Balakrishnan and N.L. Johnson. Continuous Multivariate distributions, John Wiley and sons, 2000.

MA2130: LAB ON DISTRIBUTION THEORY [0 0 2 1] The following practical will be performed using statistical software: Fitting of the Binomial distribution, Poisson distribution, Geometric distribution, Negative Binomial distribution, Multinomial distribution, Hyper-geometric distribution, Uniform distribution, Normal distribution, Exponential distribution, Beta distribution, Gamma distribution, Weilbul distribution and Cauchy distribution. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015.

2. Gopal K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY2160: ANALYTICAL CHEMISTRY [2 1 0 3] Basic Concepts: Introduction to analytical chemistry. Measurement Basics: Introduction, electrical components and circuits, operational amplifiers in chemical instrumentation. Atomic spectroscopy: Introduction to spectrometric methods, components of optical instruments, atomic absorption and atomic fluorescence spectrometry, atomic emission spectrometry, atomic mass spectrometry, atomic X-ray spectrometry. Molecular Spectroscopy: UV-Vis, IR, NMR, mass, Raman, fluorescence spectroscopy, instrumentations and applications. Electroanalytical Chemistry: Introduction to electroanalytical chemistry, potentiometry, coulometry, voltammetry, instrumentation and application. Separation Methods: An introduction to chromatographic separations, gas chromatography, high-performance liquid chromatography, capillary electrophoresis and capillary electrochromatography, components of instruments and applications. Miscellaneous Methods: Thermal methods for analytical chemistry, instrumentation and applications. References:

1. D. A. Skoog, F. J. Holler, T. A. Nieman, Principles of Instrumental Analysis, Saunders College Publishing, 2013.

2. H. H. Willard, L. L. Merritt Jr., J. A. Dean, F. A. Settle, Instrumental Methods of Analysis, CBS Publishing Company, 2012.

3. G.D. Christian, Analytical Chemistry, John Wiley, 2004. 4. D.A. Skoog, D.M. West, F.J. Holler, S.R. Crouch, Fundamentals of Analytical chemistry, Brooks/Cole, 2004.

CY2138: ANALYTICAL CHEMISTRY LABORATORY [0 0 2 1] Analytical: TLC, paper chromatography, determination of Rf values, separation techniques. Reference:

2. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book

Agency, 2011.

PY2161: HEAT AND THERMODYNAMICS [3 1 0 4] Thermodynamics: Thermodynamic equilibrium, zeroth law of thermodynamics and concept of temperature, work and heat energy, state functions. Laws of Thermodynamics: First law of thermodynamics, differential form of first law, internal energy, first law and various processes, applications of first law, heat engines, Carnot cycle, Carnot engine, second law of Thermodynamics-Kelvin-Planck and Clausius Statements and their equivalence, Carnot theorem; Applications of Second Law of Thermodynamics: Thermodynamic scale of temperature and its equivalence to perfect gas scale. Entropy: Change in entropy, entropy of a state, Clausius theorem, second law of thermodynamics in terms of entropy, entropy of a perfect gas, entropy of the universe, entropy changes in reversible and irreversible processes, principle of increase of entropy. Impossibility of Attainability of Absolute Zero: Third law of thermodynamics, temperature-entropy diagrams, first and second order phase transitions. Thermodynamic Potentials: extensive and intensive thermodynamic variables, thermodynamic potentials U, H, F and G, their definitions, properties and applications, surface films and variation of surface tension with temperature, magnetic work, cooling due to adiabatic demagnetization, approach to absolute zero. Maxwell’s Thermodynamic Relations: Derivations of Maxwell’s relations, applications of Maxwell’s relation, Clausius-Clapeyron equation, Joule-Kelvin Coefficient for Ideal and Van der Waal Gases; Kinetic Theory of Gases: Distribution of velocities, Maxwell-Boltzmann law of distribution of velocities in an ideal gas and its experimental verification, mean, RMS and most probable speeds, degrees of freedom, law of equipartition of energy, specific heats of gases, molecular collisions, mean free path, collision probability, estimates of mean free path. Transport Phenomenon in Ideal Gases: Viscosity, thermal conductivity and diffusion, Brownian motion and its significance; Real gases: Behavior of real gases, Van der Waal’s equation of state for real gases, values of critical constants, Joule’s Experiment, free adiabatic expansion of a perfect gas, Joule-Thomson Porous Plug Experiment, Joule-Thomson effect for real and van der waal gases, temperature of inversion, Joule-Thomson Cooling. References:

1. M. W. Zemansky, R. Dittman, Heat and Thermodynamics, McGraw-Hill, 2017. 2. S. C. Garg, R. M. Bansal, C. K. Ghosh, Thermal Physics: with Kinetic Theory, Thermodynamics and Statistical

Mechanics, McGraw-Hill, 2017.

3. E. Fermi, Thermodynamics, Snow Ball Publications, 2010. 4. F. W. Sears, G. L. Salinger, Thermodynamics Kinetic Theory and Statistical Thermodynamics, Narosa

Publications, 1998. 5. M. Alonso, E. Finn, Physics, Addison-Wesley, 2000.

GE – III (B) & LAB

MA2141: SAMPLING THEORY [2 1 0 3] Introduction: Concept of population and sample, need for sampling, complete enumeration versus sampling, basic concepts in sampling, sampling and non-sampling errors, acquaintance with the working (questionnaires, sampling design, methods followed in field investigation, principal findings, etc.) of NSSO and other agencies under taking sample surveys; Simple Random Sampling With or Without Replacement: Estimation of mean, total, variance, proportion, equivalence of different definitions, sample size problem; Stratified Sampling: Estimation of mean, total, proportion and optimum allocation, comparison with simple random sampling without replacement, post stratification; Systematic Sampling Scheme: Linear and circular systematic sampling, comparison with SRS and stratified sampling schemes, linear trend, simple method of variance; Cluster Sampling: Estimation of mean and total, relative efficiency and its estimation, optimum unit of sampling and multipurpose surveys, result on equal and unequal clusters; Two Stage Sampling: results on equal and unequal first sampling unit, allocation of sample, comparison with one stage sampling, effect of change in size of first sampling unit. References:

1. S. Singh, Advanced Sampling Theory and Applications, Springer Science Business Media, 2003. 2. P. Mukhopadhyay, Theory and Methods of Survey Sampling, New Delhi, 2015. 3. D. Singh and F.S. Chaudhary, Theory and Analysis of Sample Survey Designs, New Age International

Publication, 2018. 4. S. Sampath, Sampling Theory and Methods, Narosa Publishing House, New Delhi, 2000. 5. Des Raj, Sample Survey Theory, Narosa Publishing House, New Delhi, 2000. 6. W.G. Cochran, Sampling Techniques, John Wiley and Sons, New York, 1977. 7. P.V. Sukhtme, B.V. Sukhatme, S. Sukhatme, and C. Ashok, Sampling Theory of Surveys with Applications,

Indian Society of Agricultural Statistics, New Delhi, 1984. MA2131: LAB ON SAMPLING THEORY [0 0 2 1] The following practical will be performed using statistical software: Estimation of population mean and variance of estimator under simple random sampling, systematic sampling stratified random sampling, cluster sampling and two stage sampling. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. G. K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY2161: STRUCTURE OF MATERIALS [2 1 0 3] Basic Concepts: Introduction to inorganic chemistry. Structure of crystalline solids: Classification of materials, crystalline and amorphous solids crystal. Structure, symmetry and point groups, Brvais lattice, unit cells, types of close packing - hcp and ccp, packing efficiency, radius ratios; crystallographic direction and plane. Ceramics: Classification, structure, impurities in solids. Electrical Properties: Introduction, basic concept of electric conduction, free electron and band theory, classification of materials, insulator, semiconductor, intrinsic & extrinsic semi-conductors, metal, superconductor etc., novel materials. Magnetic Properties: Introduction, origin of magnetism, units, types of magnetic ordering: dia-para-ferro-ferri and antiferro-magnetism, soft and hard magnetic materials, examples of some magnetic materials with applications. Special topics: Biomaterials, nanomaterials, composite materials. References:

1. W. D. Callister, Material Science and Engineering, An introduction, 3rd Edition, Willey India, 2009.

2. H. V. Keer, Principals of Solid State, Willey Eastorn, 2011. 3. J. C. Anderson, K. D. Leaver, J. M. Alexander, & R. D. Rawlings, Materials Science, Willey India, 2013.

CY2139: MATERIAL CHEMISTRY LABORATORY [0 0 2 1] Materials: Quantitative estimation of mixtures. Reference:

1. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book Agency, 2011.

PY2160: ELECTRONICS [3 1 0 4] Network theorems: Fundamentals of AC and DC networks, Thevenin, Norton, superposition, maximum power transfer theorem; Semiconductor Diodes: P and N type semiconductors, energy level diagram, conductivity and mobility, drift velocity, p-n junction, barrier formation in p-n junction diode, static and dynamic resistance. Two-terminal Devices and their Applications: Rectifiers, half wave, full wave and bridge, ripple factor, zener diode and voltage regulation, principle and structure of LEDs, photodiode, tunnel diode, solar cell. Bipolar Junction transistors: n-p-n and p-n-p transistors, characteristics of CB, CE and CC configurations, current gains α and β, relations between α and β, load line analysis, DC load line and Q-point, active, cutoff, saturation region. Amplifiers: Transistor biasing and stabilization circuits, fixed

Bias and voltage divider bias, transistor as 2-port network, h-parameter, equivalent circuit, analysis of a single-stage CE amplifier using hybrid model, input and output Impedance, current, voltage and power gain, class A, B & C amplifiers, coupled amplifier, RC-coupled amplifier and its frequency response. Feedback in Amplifiers: Effects of positive and negative feedback on input impedance, output impedance, gain, stability, distortion and noise; Sinusoidal Oscillators: Barkhausen's criterion for self-sustained oscillations. RC Phase shift oscillator, determination of frequency, Hartley & Colpitts oscillators; Operational Amplifiers and its Applications: characteristics of an ideal and practical Op-Amp, open-loop and closed-loop gain, frequency response, CMRR, slew rate and concept of virtual ground, inverting and non-inverting amplifiers, adder, subtractor, differentiator, integrator, log amplifier, zero crossing detector Wein bridge oscillator. Three-terminal Devices (UJT and FETs): Characteristics and equivalent circuit of UJT and JFET, advantages of JFET, MOSFET. References:

1. B. G. Streetman, S. Banerjee, Solid state electronic devices, Pearson Prentice Hall, 2015. 2. R. Boylestad, Louis Nashelsky, Electronic Devices and Circuit Theory, Pearson Education, India, 2014. 3. pA. B. Gupta, N Islam, Solid State Physics and Electronics, Books & Allied Ltd, 2012 4. D Chattopadhyay, P C Rakshit, Electronics: Fundamentals and Applications, New Age international (P) Ltd,

2018. 5. J. Millman, C. C. Halkias, Integrated Electronics, Tata McGraw-Hill, 2017 6. pA. P. Malvino, Electronic Principals, McGraw-Hill, 2015. 7. Mottershead, Electronic Circuits and Devices, PHI, 1997. 8. N. N. Bhargava, D. C. Kulshreshtha, S. C. Gupta, Basic Electronics and Linear Circuits, Tata Mc-Graw-Hill,

2012.

GE – IV & LAB MA3240: DESIGN OF EXPERIMENTS [2 1 0 3] Analysis of Variance: Analysis of Variance for one- way, two -way with one/m observations per cell for fixed, mixed and random effects models, Tukey’s test for non- additivity; Design of Experiment: Basic principles of experimental design, general block design and its information matrix, criteria of connectedness, balance and orthogonality, analysis of completely randomized, randomized blocks and Latin-square designs; Factorial Experiments: Symmetrical factorials,

22-experiment and 23-experiment. References:

1. M.N. Das and N.C. Giri, Design and Analysis of Experiments, Wiley Eastern Ltd, 1986. 2. A.M. Goon, M.K. Gupta and B. Dasgupta, Fundamentals of Statistics, Vol. II, 8th edition. World Press, Kolkata,

2005. 3. S.C. Gupta and V.K. Kapoor, Fundamentals of Applied Statistics, Sultan Chand and Co., 3rd edition, New Delhi,

2008. 4. M. D. Morris, Design of Experiment An Introduction Based On Linear Models, Chapman and Hall/CRS, 2017. 5. D. C. Montgomery, Design and Analysis of Experiments, John Wiley, 2008. 6. A. Dey, Theory of Block Design, J. Wiley, 1986.

MA3230: LAB ON DESIGN OF EXPERIMENTS [0 0 2 1] The following practical will be performed using statistical software: Analysis of for one- way, two -way with one/m observations per cell for fixed, mixed and random effects models, analysis of completely randomized, randomized

blocks and Latin-square designs, analysis of 22-experiment, 23-experiment and 2𝑛-experiment in 2𝑘 blocks per replicate. References:

1. M. J. Crawley, Statistics: An Introduction Using R, Wiley, 2015. 2. Gopal K. Kanji, 100 Statistical Tests, SAGE Publication, 3rd edition, 2006.

CY3260: BIOPHYSICAL CHEMISTRY [2 1 0 3] Basic Concepts: Introduction to physical chemistry. General Biophysical Principles: Laws of biophysics, hydrogen bonding, van der Waals and hydrophobic interactions, disulphide bridges, role of water and weak interactions, energies, forces & bonds, kinetics of biological processes, electron transport & oxidative phosphorylation. Methods in Biophysics: Analytical ultracentrifugation, micro calorimetry, x-ray diffraction, spectroscopy – UV, IR, NMR, mass fluorescence, circular dichroism, microscopy, separation techniques. Molecular Biophysics: Principles of protein structure & confirmation, proteins structure and stability, structure of nucleic acids. Protein Engineering: Micro sequencing methods for proteins & engineering proteins for purification chemical approach to protein engineering & protein engineering for thermostability. Membrane Biophysics: Membrane structure & models, physical properties of membrane, membrane transport, molecular dynamics of membranes, Membrane potential and lipid membrane technology. References:

1. D. L. Nelson, M. M. Cox, Lehninger’s Principles of Biochemistry, W. H. Freeman, 2015. 2. Satyanarayana, Biochemistry, Elsevier, 2017. 3. J. M. Berg, J. L. Tymoczko, L. Stryer, Biochemistry, W. H. Freeman, 2011.

CY3238: APPLIED CHEMISTRY LABORATORY [0 0 2 1] Applied chemistry: Water analysis, effluent analysis, pH-metric and conductometric titrations. Computational: Scientific software, data handling. Reference:

1. A. K. Nad, B. Mahapatra, & A. Ghoshal, An Advanced Course in Practical Chemistry, New Central Book Agency, 2011.

PY3260: MODERN PHYSICS [3 1 0 4] Particles and Waves: Inadequacies in classical physics, blackbody radiation, photoelectric effect, Compton Effect, Franck-Hertz experiment, wave nature of matter, wave packets, group and phase velocities, two-slit experiment with electrons, probability, wave functions, Heisenberg’s uncertainty principle, derivation from wave packets, γ-ray microscope. quantum mechanics: basic postulates and formalism: energy, momentum and Hamiltonian Operators, time-independent Schrödinger wave equation for stationary states, conditions for physical acceptability of wave functions, expectation values, wave function of a free particle. Applications of Schrödinger Wave Equation: Eigen functions and eigenvalues for a particle in a one dimensional box: bound state problems, general features of a bound particle system, (1) one dimensional simple harmonic oscillator, scattering problems in one dimension: (1) finite potential step: reflection and transmission, stationary solutions, probability current, attractive and repulsive potential barriers (2) quantum phenomenon of tunneling: tunnel effect, tunnel diode (qualitative description) (3) finite potential well (square well). Operators in Quantum Mechanics: Hermitian operator, commutator brackets, simultaneous eigen functions, commutator algebra, commutator brackets using position, momentum and angular momentum operator, concept of parity, parity operator and its eigen values. References:

1. A. Ghatak, S. Lokanathan, Quantum Mechanics: Theory and Applications, Laxmi Publications, 2016. 2. D. J. Griffith, Introduction to Quantum Mechanics, Pearson Education, 2015. 3. L. I. Schiff, J. Bandhyopadhyay, Quantum Mechanics, McGraw-Hill Book, 2010. 4. E. Merzbacher, Quantum Mechanics, John Wiley & Sons, Inc, 2007. 5. J. L. Powell, B. Crasemann, Quantum Mechanics, Addison-Wesley Pubs.Co., 2010. 6. E. M. Lifshitz, L. D. Landau, Quantum Mechanics: Non-Relativistic Theory, Butterworth-Heinemann, 2009.

M. Sc. Mathematics

Year

FIRST SEMESTER SECOND SEMESTER

Course Code

Course Name L T P C Course Code

Course Name L T P C

I

MA6111 Advanced Linear Algebra 3 1 0 4 MA6201 Partial Differential Equations 3 1 0 4

MA6112 Mathematical Analysis 3 1 0 4 MA6202 Optimization Theory and Techniques 3 1 0 4

MA6113 Differential Equations 2 1 0 3 MA6203 Functional Analysis 3 1 0 4

MA6114 Advanced Complex Analysis 3 1 0 4 MA6204 Measure theory & Integration 3 1 0 4

MA6115 Mathematical Statistics 3 1 0 4 MA6205 Research Methodology & Technical Writing 3 1 0 3

MA6116 Topology-I 2 1 0 3 MA6230 Lab on Optimization Theory and Techniques 0 0 2 1

MA6270 Seminar 0 0 0 1

16 6 0 22 15 5 2 21

Total Contact Hours (L + T + P) 22 Total Contact Hours (L + T + P) 22

II

THIRD SEMESTER FOURTH SEMESTER

MA7101 Fluid Dynamics 3 1 0 4

MA7270 Project 0 0 0 16

MA7102 Special Function & Integral Transformation 2 1 0 3

MA7103 Integral Equations and Calculus of Variations 3 1 0 4

MA7104 Theory of Field Extensions 3 1 0 4

****** DSE – I 2 1 0 3

****** DSE – II 2 1 0 3

15 6 0 21 0 0 0 16

Total Contact Hours (L + T + P) 21 Total Contact Hours (L + T + P) + OE 0

Discipline Specific Elective (DSE)

DSE-I 1. MA7140: Mechanics of Solids 2. MA7141: Stochastic Process 3. MA7142: Fuzzy Sets & Their Applications

DSE-II 1. MA7143: Topology-II 2. MA7144: Linear Models 3. MA7145: Computational Fluid Dynamics

MA6111: ADVANCED LINEAR ALGEBRA [3 1 0 4] Linear Transformations: Recall of vector space, basis, dimension and related properties, algebra of linear transformations, vector space of linear transformations L(U,V), dimension of space of linear transformations, change of basis and transition matrices, linear functional, dual basis, computing of a dual basis, dual vector spaces, annihilator, second dual space, dual transformations; Inner-Product Spaces: Normed space, Cauchy-Schwartz inequality, pythagorean theorem, projections, orthogonal projections, orthogonal complements, orthonormality, matrix representation of inner-products, Gram-Schmidt orthonormalization process, Bessel’s inequality, Riesz representation theorem and orthogonal transformation, Inner product space isomorphism, operators on inner-product spaces, isometry on inner-product spaces and related theorems, adjoint operator, selfadjoint operator, normal operator and their properties, matrix of adjoint operator , algebra of Hom (V,V), minimal polynomial, invertible linear transformation, characteristic roots, characteristic polynomial and related results; Diagonalization: Diagonalization of matrices, invariant subspaces, Cayley-Hamilton theorem, canonical form, Jordan Form. Forms on vector spaces, bilinear functionals, symmetric bilinear forms, skew symmetric bilinear forms, rank of bilinear forms, quadratic forms, and classification of real quadratic forms. References:

1. K. B. Datta, Matrix and Linear Algebra, Prentice Hall of India Pvt. Ltd, New Delhi, 2007. 2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, First course in Linear Algebra, New Age International Ltd,

2012 3. K. Hoffman and R. Kunze, Linear Algebra, 2nd edition, Prentice Hall, Englewood Cliffs, New Jersey, 2014. 4. S. Kumaresan, Linear Algebra-A geometric approach, Prentice Hall of India, 5. 2000. 6. R. B. Dash and D. K. Dalai, Fundamentals of Linear Algebra, Himalaya Publishing house, 2008. 7. S. Lang, Linear Algebra, 3rd edition, Springer-Verlag, New York 2005.

MA6112: MATHEMATICAL ANALYSIS [3 1 0 4] Riemann-Stieltjes Integral: Introduction, existence and properties, integration and differentiation, fundamental theorem of calculus, integration of vector-valued functions, rectifiable curves; Sequence and Series of Functions: Pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weirstrass M test, Abel and Dirichlet tests for uniform convergence, uniform convergence and continuity, uniform convergence and differentiation, Weierstrass approximation theorem, power series, uniform convergence and uniqueness theorem, Abel theorem, Tauber theorem; Functions of Several Variables: Linear transformations, Euclidean space Rn, derivatives in an open subset of Rn, chain rule, partial derivatives, continuously differentiable mapping, Young and Schwarz theorems, Taylor theorem, higher order differentials, explicit and implicit functions, implicit function theorem, inverse function theorem, change of variables, extreme values of explicit functions, stationary values of implicit functions, Lagrange multipliers method, Jacobian and its properties. References:

1. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, Kogakusha, 2017. 2. H.L. Royden, Real Analysis, Macmillan Pub. Co., Inc. 4th edition, New York, 2009. 3. S.C. Malik and Savita Arora, Mathematical Analysis, New Age International Limited, New Delhi, 2012. 4. T. M. Apostol, Mathematical Analysis, Addison-Wesley Publishing Company, 2008. 5. G. De Barra, Measure Theory and Integration, Wiley Eastern Limited, 2003. 6. R. G. Bartle, The Elements of Real Analysis, Wiley International Edition, 2011.

MA6113: DIFFERENTIAL EQUATIONS [2 1 0 3] Preliminaries: ε-approximate solution, Cauchy-Euler construction of an ε-approximate solution of an initial value problem, Equicontinuous family of functions; Basic Theorems: Ascoli-Arzela lemma, Cauchy-Peano existence theorem, Lipschitz condition, Picards-Lindelof existence and uniqueness theorem for dy/dt=f(t,y), Solution of initial-value problems by picards method; Dependence of Solutions on Initial Conditions: Linear systems, Matrix method for homogeneous first order system of linear differential equations; Fundamental Set of Solutions: Fundamental matrix of solutions, Wronskian of solutions, basic theory of the homogeneous linear system, Abel-Liouville formula, nonhomogeneous linear system. Strum theory, self-adjoint equations of the second order, Abel formula, Strum separation theorem, Strum fundamental comparison theorem, nonlinear differential systems, phase plane, path, critical points; Poincore- Bendixson Theory: Autonomous systems, isolated critical points, path approaching a critical point, Path entering a critical point, types of critical points, enter, saddle points, spiral points, node points, stability of critical points, Asymptotically stable points, unstable points, critical points and paths of linear systems, almost linear systems, nonlinear conservative dynamical system, dependence on a parameter, Liapunov direct method, limit cycles, periodic solutions, Bendixson nonexistence criterion, poincore- Bendixson theorem, index of a critical point, Strum-Liouville problems, orthogonality of characteristic functions. References:

1. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw Hill, 2000. 2. S.L. Ross, Differential Equations, John Wiley and Sons Inc., New York, 2004. 3. W.E. Boyce and R.C. Diprima, Elementary Differential Equations and Boundary Value Problems, John Wiley

and Sons, Inc., New York, 4th edition, 2012. 4. G.F. Simmon, Differential Equations, Tata McGraw Hill, New Delhi, 2016.

MA6114: ADVANCED COMPLEX ANALYSIS [3 1 0 4]

Integral Functions: Factorization of an integral function, Weierstrass primary factors, Weierstrass’ factorization theorem, Gamma function and its properties, Stirling formula integral version of gamma function, Riemann Zeta function, Riemann functional equation, Mittag-Leffler theorem, Runge theorem; Analytic Continuation: Natural boundary, uniqueness of direct analytic continuation, uniqueness of analytic continuation along a curve, power series method of analytic continuation, Schwarz reflection principle, germ of an analytic function, monodromy theorem and its consequences, Harmonic functions on a disk, Poisson kernel, Dirichlet problem for a unit disc, Harnack inequality, Harnack theorem, Dirichlet region, Green function, Canonical product, Jensen formula, Poisson-Jensen formula, Hadamard three circles theorem; Entire Function: Growth and order of an entire function, an estimate of number of zeros, exponent of convergence, Borel theorem, Hadamard factorization theorem, range of an analytic function, Bloch theorem, Schottky theorem, Little Picard theorem, Montel Caratheodory theorem, Great Picard theorem, univalent functions, Bieberbach conjecture and the “1/4 theorem” . References:

1. S. Ponnusamy, Foundations of Complex Analysis, Narosa Publishing House, 2011. 2. J.B. Conway, Functions of one Complex variable, Springer-Verlag, Narosa Publishing House, 2002. 3. H.S. Kasana, Complex Variable Theory and Applications, PHI Learning Private Ltd, 2011. 4. M. J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press,

South Asian Edition, 2003. 5. R. V. Churchill and James Ward Brown, Complex Variables and Applications, McGraw-Hill Publishing

Company, 2013. 6. L.V. Ahlfors, Complex Analysis, Mc-Graw Hill, 1979.

MA6115: MATHEMATICAL STATISTICS [3 1 0 4] Probability: Definition and various approaches of probability, addition theorem, Boole inequality, conditional probability and multiplication theorem, independent events, mutual and pairwise independence of events, Bayes theorem and its applications; Random Variable and Probability Functions: Definition and properties of random variables, discrete and continuous random variables, probability mass and density functions, distribution function, concepts of bivariate random variable: joint, marginal and conditional distributions, cumulative generating function; Mathematical Expectation: Definition and its properties. variance, covariance, moment generating function- Definitions and their properties; Discrete Distributions: Uniform, Bernoulli, Binomial, Poisson and Geometric distributions with their properties; Continuous Distributions: Uniform, Normal, Exponential, Beta and Gamma distributions with their properties; Testing of Hypothesis: Parameter and statistic, sampling distribution and standard error of estimate, null and alternative hypotheses, simple and composite hypotheses, critical region, Level of significance, one tailed and two tailed tests, two types of errors; Tests of Significance: Large sample tests for single mean, single proportion, difference between two means and two proportions. References:

1. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand and Co., 3rd edition, New Delhi, 2008.

2. V.K. Rohtagi and A.K.M. E Saleh, An Introduction to Probability & Statistics, John Wiley & Sons, 2011. 3. P. L. Meyer, Introductory Probability and Statistical Applications, Addison-Wesley, 2017. 4. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edition, John Wiley, 2005. 5. P. Mukhopadhyay, Mathematical Statistics, Books & Allied (P) Ltd., 2009. 6. G. Casella, and R.L. Berger, Statistical Inference, 2nd edition. Thomson Duxbury, 2002. 7. R.V. Hogg, and E.A. Tanis, Probability and Statistical Inference, 9th edition, Macmillan Publishing Co. Inc.,

2014. MA6116: TOPOLOGY-I [2 1 0 3] Basic Concepts: Definition and examples of topological spaces, comparison of topologies on a set, intersection and union of topologies on a set, neighborhoods, Interior point and interior of a set , closed set as a complement of an open set , adherent point and limit point of a set, closure of a set, derived set, properties of closure operator, boundary of a set , dense subsets, interior, exterior and boundary operators, alternative methods of defining a topology in terms of neighborhood system and Kuratowski closure operator, relative (induced) topology, base and subbase for a topology, Base for Neighborhood system, continuous functions, open and closed functions, homeomorphism. connectedness and its characterization; Connected Spaces: connected subsets and their properties, continuity and connectedness, components, locally connected spaces; Compact Spaces : Compact spaces and subsets, compactness in terms of finite intersection property, continuity and compact sets, basic properties of compactness, closeness of compact subset and a continuous map from a compact space into a Hausdorff and its consequence, sequentially and countably compact sets, Local compactness and one point compatification; Seperations Axioms: First countable, second countable and separable spaces, Hereditary and topological property, countability of a collection of disjoint open sets in separable and second countable spaces, Lindelof theorem, T0, T1, T2 (Hausdorff) separation axioms, their characterization and basic properties.

References: 1. C.W. Patty, Foundation of Topology, Jones & Bertlett, 2009. 2. Fred H. Croom, Principles of Topology, Cengage Learning, 2009. 3. G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, 1983. 4. J. R. Munkres, Toplogy, Pearson Education Asia, 2002. 5. K. Chandrasekhara Rao, Topology, Narosa Publishing House Delhi, 2009. 6. K.D. Joshi, Introduction to General Topology, Wiley Eastern Ltd, 2006.

7. W.J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 2014.

MA6201: PARTIAL DIFFERENTIAL EQUATIONS [3 1 0 4] Partial Differential Equations(PDE): 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, 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, separation of variables in a PDE, Higher Order Partial Differential Equations: Laplace’s equation, elementary solutions of Laplace’s equations, families of equipotential surfaces, wave equation, the occurrence of wave equations, elementary solutions of one dimensional wave equation, diffusion equation, resolution of boundary value problems for diffusion equation, elementary solutions of diffusion equation, separation of variables. References:

1. I.N. Sneddon, Elements of Partial Differential Equation, 3rd edition, Dove Publication, 2006. 2. M.D. Raisinghania, Ordinary and Partial Differential Equations, S. Chand & Sons, 2010. 3. E.T. Copson, Partial Differential Equations, Cambridge University Press, 1995. 4. L.C. Evans, Partial Differential Equations, Vol. 19, AMS, 2010. 5. J.R. Buchanan and Z. Shao, A First Course of Partial Differential Equation, World Scientific Publishing, 2017.

MA6202: OPTIMIZATION THEORY AND TECHNIQUES [3 1 0 4] Unconstrained Optimization: Fibonacci golden section and quadratic interpolation methods for one dimensional problems, steepest descent, conjugate gradient and variable metric methods for multidimensional problems; Nonlinear Programming: Generalized convexity, quasi and psuedo convex functions and their properties, general nonlinear programming problem, difficulties introduced by nonlinearity, Kuhun-Tucker necessary conditions for optimality, insufficiency of K-T conditions, sufficiency conditions for optimality, solution of simple NLPP using K-T conditions; Quadratic Programming: Beale’s method, restricted basis entry method (Wolfe’s method), proof of termination for the definite case, resolution of the semi definite case, duality in quadratic programming; Convex Programming: Methods of feasible directions, Zoutendijk’s method, Rozen’s gradient projection method for linear constraints, Kelly’s cutting plane method to deal with nonlinear constraints. References:

1. S.S. Rao, Optimization Theory and Applications, Wiley Eastern, 2009. 2. G. Hadley, Nonlinear and Dynamic Programming, Addison Wesley, 2018. 3. M. Bazara and Shetty, Nonlinear Programming: Theory and Algorithms, 3rd edition, John Wiley, 2006. 4. H.S. Kasana, Introductory Operation Research: Theory and Applications, Springer Verlag, 2005. 5. R. L. Rardin, Optimization in Operations research, Pearson Education, 2005.

MA6203: FUNCTIONAL ANALYSIS [3 1 0 4] Normed Linear Spaces: Metric on normed linear spaces, completion of a normed space; Banach Spaces: Introduction, subspace of a banach space, Holder and Minkowski inequality, completeness of quotient spaces of normed linear spaces, completeness of lp, Lp, Rn, Cn and C[a,b], incomplete normed spaces, finite dimensional normed linear spaces and subspaces, bounded linear transformation, equivalent formulation of continuity, spaces of bounded linear transformations, continuous linear functional, conjugate spaces, Hahn-Banach extension theorem (real and complex form), Riesz representation theorem for bounded linear functionals on Lp and C[a,b], second conjugate spaces, reflexive space, uniform boundedness principle and its consequences, open mapping theorem and its application, projections, closed graph theorem equivalent norms, weak and strong convergence, their equivalence in finite dimensional spaces, weak sequential compactness, solvability of linear equations in banach spaces; Compact Operator Theory: Compact operator and its relation with continuous operator, compactness of linear transformation on a finite dimensional space, properties of compact operators, compactness of the limit of the sequence of compact operators. References:

1. G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, 2003. 2. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley, 2007. 3. A. H. Siddiqi, Khalil Ahmad and P. Manchanda, Introduction to Functional Analysis with Applications, Anamaya

Publishers, New Delhi, 2006 4. K.C. Rao, Functional Analysis, Narosa Publishing House, 2nd edition, 2006

MA6204: MEASURE THEORY & INTEGRATION [3 1 0 4] Measurable Sets: Set functions, intuitive idea of measure, elementary properties of measure, measurable sets and their fundamental properties, Lebesgue measure of a set of real numbers, algebra of measurable sets, Borel set, equivalent formulation of measurable sets in terms of open, closed, non-measurable sets, measurable functions and their equivalent formulations, properties of measurable functions, approximation of a measurable function by a sequence of simple functions, measurable functions as nearly continuous functions, Egoroff theorem, Lusin theorem, convergence in measure and F. Riesz theorem, almost uniform convergence; Measureable Function and Lebesgue Integral: Shortcomings of Riemann integral, Lebesgue integral of a bounded function over a set of finite measure and its properties, Lebesgue integral as a generalization of Riemann integral, bounded convergence theorem, Lebesgue theorem regarding points of discontinuities of Riemann integrable functions, integral of non-negative functions, Fatou

lemma, monotone convergence theorem, general Lebesgue integral, Lebesgue convergence theorem, Vitali covering lemma, differentiation of monotonic functions, function of bounded variation and its representation as difference of monotonic functions, differentiation of indefinite integral, Fundamental theorem of calculus, absolutely continuous functions and their properties. References:

1. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, Kogakusha, 2017. 2. H.L. Royden, Real Analysis, Macmillan Pub. Co., Inc. 4th edition, New York, 1993. 3. P. K. Jain and V. P. Gupta, Lebesgue Measure and Integration, New Age International (P) Limited Published,

New Delhi, 2012. 4. G. De Barra, Measure Theory and Integration, Wiley Eastern Ltd., 2003. 5. R.R. Goldberg, Methods of Real Analysis, Oxford & IBH Pub. Co. Pvt. Ltd, 2012. 6. R. G. Bartle, The Elements of Real Analysis, Wiley International Edition, 2011. 7. R. R. Goldberg, Methods of Real Analysis, John Wiley & Sons, 2012.

MA6205: RESEARCH METHODOLOGY & TECHNICAL WRITING [2 1 0 3] Foundations of Research: Meaning, objectives, motivation, utility, empiricism, deductive and inductive theory, characteristics of scientific method , understanding the language of research; Research Process: Problem identification & formulation, research question, investigation question, measurement issues, hypothesis, qualities of a good hypothesis, types of hypothesis; Research Design: Concept and importance in research, features of a good research design, exploratory research design, descriptive research designs, experimental research design; Types of Data: Classification of data, uses, advantages, disadvantages, sources; Measurement: Concept of measurement, problems in measurement in research, validity and reliability, levels of measurement; Statistical Techniques and Tools: Introduction of statistics, functions, limitations, graphical representation, measures of central tendency, measure of dispersion, skewness, kurtosis, correlation, regression, tests of significance based on t, F, Chi-square, Z and ANOVA test; Paper Writing: Layout of a research paper, Scopus/Web of Science journals, impact factor of journals, when and where to publish, ethical issues related to publishing, plagiarism and self-plagiarism. Introduction to LATEX and MATLAB. References:

1. C.R. Kothari, Research Methodology Methods & Techniques, New Age International Publishers, Reprint 2008. 2. R. Singh, Research Methodology, Saga Publication, 4th edition, 2014. 3. J. Anderson and M. Poole, Thesis and Assignment Writing, Wiley India 4th edition, 2011. 4. Mukul Gupta and Deepa Gupta, Research Methodology, PHI Learning Private Ltd., New Delhi, 2011. 5. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, New Delhi,

1999. MA6230: LAB ON OPTIMIZATION THEORY AND TECHNIQUES [0 0 2 1] The following practical will be performed using software: Fibonacci golden section and quadratic interpolation methods for one dimensional problems, Kuhun-Tucker necessary conditions for optimality, solution of simple NLPP using K-T conditions, Beale’s method, restricted basis entry method (Wolfe’s method), duality in quadratic programming, Methods of feasible directions, Zoutendijk’s method, Rozen’s gradient projection method for linear constraints, Kelly’s cutting plane method to deal with nonlinear constraints. Reference:

1. M.W. Carter and Camille C, Operation Research: A Practical Introduction, CRC Press, 1st edition, 2000.

MA7101: FLUID DYNAMICS [3 1 0 4] Kinematics: Euler's equations of motion, Lagrange's equations of motion, Lagrangian and Eulerian methods, equations of continuity in Lagrangian and Eulerian methods, stream line, velocity potential, path line, velocity and circulation, boundary surface, rotational and irrotational motion, equation of energy; Motion in Two Dimensions: stream function, complex velocity potential, source, sink and doublet, their image, images in two dimensions, images of a source with regard to a plane, a circle and a sphere, image of a doublet, Milne-Thomson circle theorem, theorem of Blasius; Vortex Motion: Helmholtz properties of vortices, velocity in a vortex field, motion due to circular vortex, infinite rows of vortices, Ka'rma'n vortex street; Viscous Fluid: Navier- Stokes equations; diffusion of vorticity, dissipation of energy, steady motion of a viscous fluid between two parallel planes, steady flow through cylindrical pipes. References:

1. Y. A. Cengel and John M. Cimbara, Fluid Mechanics: Fundamentals and Applications, McGraw Hill Eduction, 4th edition, 2017.

2. M.D. Raisinghania, Fluid dynamics, S. Chand Publication, 2010. 3. J.L. Bansal, Viscous Fluid Dynamics, Oxford Publications, 2003. 4. G.K. Batchelor, An Introduction to Fluid Dynamics, Foundation Books, 2005. 5. T. F. Chorlton, and Van Nostrand Reinhold, Text Book of Fluid Dynamics, Co., London, 1990.

MA7102: SPECIAL FUNCTION & INTEGRAL TRANSFORMATION [2 1 0 3] Gauss Hypergeometric Function: Introduction and its properties, Series solution of Gauss hypergeometric equation. Integral representation, Linear and quadratic transformation formulas, Contiguous function relations, Differentiation formulae, Linear relation between the solutions of Gauss hypergeometric equation, Kummer’s confluent hypergeometric function and its properties, Integral representation, Kummer’s first transformation; Legendre

Polynomials: Introduction, Rodrigue’s formula, orthogonality, recurrence relations. Functions Pn(x) and Q

n(x) and their

properties; Bessel Function: Introduction, Jn(x) and its properties, recurrence relations; Laplace Transform: Definition

and its properties. Laplace transform of Periodic functions. Properties of inverse Laplace transform. Convolution theorem. Application of Laplace Transform in Solving definite integrals, ordinary and Partial differential equations; Fourier Transform: Definition and properties of Fourier sine, cosine and complex Fourier transforms. Inversion theorems, Convolution theorem. Fourier transform of ordinary and partial derivatives. Application of Fourier Transform to differential equations; Mellin Transform: Definition and its properties. Mellin transform of derivatives and Integrals. Convolution Theorem. Applications of Mellin Transform. References:

1. G.E. Andrews, R. Askey and R. Rose, Special Functions, Cambridge University Press, 2001. 2. Vasishtha and Gupta, Integral Transforms, Krishna Prakashan Mandir, 2014. 3. R.Y. Denis and U.P. Singh, Special Function and Their Applications, Dominant Publishers, 2001. 4. N. Saran, S. D. Sharma & T. N. Trivedi, Special Functions: For Mathematical, Physical & Engineering Sciences,

Pragati Prakashan, 2008. 5. B. Davies; Integral Transforms and their Applications, Springer Science, 2013.

MA7103: INTEGRAL EQUATIONS & CALCULUS OF VARIATION [3 1 0 4] Linear Integral Equations: Some basic identities, initial value problems reduced to Volterra integral equations, methods of successive substitution and successive approximation to solve Volterra integral equations of second kind, iterated kernels and Neumann series for Volterra equations, resolvent kernel as a series, Laplace transform method for a difference kernel, solution of a Volterra integral equation of the first kind, boundary value problems reduced to Fredholm integral equations, methods of successive approximation and successive substitution to solve Fredholm equations of second kind, iterated kernels and Neumann series for Fredholm equations, resolvent kernel as a sum of series, Fredholm resolvent kernel as a ratio of two series, Fredholm equations with separable kernels, approximation of a kernel by a separable kernel, Fredholm Alternative, non homonogenous Fredholm equations with degenerate kernels, Green function, use of method of variation of parameters to construct the Green function for a nonhomogeneous linear second order boundary value problem, basic four properties of the Green function, alternate procedure for construction of the Green function by using its basic four properties, reduction of a boundary value problem to a Fredholm integral equation with kernel as Green function, Hilbert-Schmidt theory for symmetric kernels; Calculus of Variation: Motivating problems of calculus of variations, shortest distance, minimum surface of resolution, Brachistochrone problem, isoperimetric problem, Geodesic, fundamental lemma of calculus of variations, Euler equation for one dependent function and its generalization to 'n' dependent functions and to higher order derivatives, conditional extremum under geometric constraints and under integral constraint. References:

1. A.J. Jerri, Introduction to Integral Equations with Applications, Wiley-Interscience Publication, 1999. 2. R.P. Kanwal, Linear Integral Equations, Theory and Techniques, Academic Press, New York, 1996. 3. P.C. Bhakta, Integral Transforms, Integral Equations and Calculus of Variations, Sarat, 2011. 4. A.S. Gupta, Calculus of Variations with Applications, PHI Learning, 2015. 5. F.B. Hilderbrand, Methods of Applied Mathematics, Dover Publications, 2005. 6. I.M. Gelfand, S.V. Fomin, Calculus of Variations, Dover Publications, 2000.

MA7104: THEORY OF FIELD EXTENSIONS [3 1 0 4] Extension of Fields: Elementary properties, simple extensions, algebraic and transcendental extensions, factorization of polynomials, splitting fields, algebraically closed fields, separable extensions, perfect fields; Galios Theory: Automorphism of fields, monomorphisms and their linear independence, fixed fields, normal extensions, normal closure of an extension, fundamental theorem of Galois theory, Norms and traces, normal basis, Galios fields, cyclotomic extensions, cyclotomic polynomials, cyclotomic extensions of rational number field, cyclic extension, Wedderburn theorem, ruler and compasses construction, solutions by radicals, extension by radicals, generic polynomial, algebraically independent sets, insolvability of the general polynomial of degree n ≥ 5 by radicals. References:

1. I.S. Luther and I.B.S. Passi, Algebra, Vol. IV-Field Theory, Narosa Publishing House, 2012. 2. I. Stewart, Galios Theory, Chapman and Hall/CRC, 2004. 3. V. Sahai and V. Bist, Algebra, Narosa Publishing House, 2003. 4. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University

Press, Indian Edition, 2012. 5. S. Lang, Algebra, 3rd edition, Addison-Wesley, 2002. 6. I. T. Adamson, Introduction to Field Theory, Cambridge University Press, 2007.

DSE - I

MA7140: MECHANICS OF SOLIDS [2 1 0 3] Tensor Analysis: Cartesian tensors of different orders, contraction of a tensor, multiplication and quotient laws for tensors, substitution and alternate tensors, symmetric and skew symmetric tensors, isotropic tensors, eigenvalues and eigenvectors of a second order symmetric tensor; Analysis of Stress: Stress vector, normal stress, shear stress, stress components, Cauchy equations of equilibrium, stress tensor of order two, symmetry of stress tensor, stress quadric of Cauchy, principal stresses, stress invariants, maximum normal and shear stresses, mohr diagram; Analysis of Strain:

Affine transformations, infinitesimal affine deformation, pure deformation, components of strain tensor and their geometrical meanings, strain quadric of Cauchy, principal strains, strain invariants, general infinitesimal deformation, saint-venant conditions of compatibility, finite deformations; Equations of Elasticity: Generalized Hook's law, Hook's law in an elastic media with one plane of symmetry, orthotropic and transversely isotropic symmetries, homogeneous isotropic elastic media, elastic moduli for an isotropic media, equilibrium and dynamical equations for an isotropic elastic media, Beltrami - Michell compatibility conditions. References:

1. M. Teodar Atanackovic and Ardeshiv Guran, Theory of Elasticity for Scientists and Engineers, Birkhausev, Boston, 2000.

2. A. K. Singh, Mechanics of Solid, Prentice Hall India Learning Private Limited, 2007. 3. A.S. Saada., Elasticity-Theory and applications, Pergamon Press, New York, 2009. 4. D.S. Chandersekhariah and L. Debnath, Continuum Mechanics, Academic Press, 1994. 5. A.K. Malik and S.J. Singh, Deformation of Elastic Solids, Prentice Hall, New Jersey, 1991

MA7141: STOCHASTIC PROCESS [2 1 0 3] Probability Generating Functions: Introduction, probability generating function, mean, variance, sum of random variables, stochastic sum, generating function of bivariate distribution, Laplace transforms and its properties, Laplace transform of a probability distribution or of a random variable, mean and variance in terms of Laplace transform, three important theorems, randomization and mixtures and classification of distributions; Stochastic Processes: Introduction, definition and examples of stochastic process, classification of general stochastic processes into discrete/continuous time, discrete/continuous state spaces, types of stochastic processes elementary problems, random walk, gambler's ruin problem; Markov Chains: Definition and examples of Markov chain, transition probability matrix, classification of states, recurrence, simple problems, basic limit theorem of Markov chain, stationary probability distribution, applications; Continuous Time Markov Chain: Poisson process and related inter-arrival time distribution, pure birth process, pure death process, birth and death process, problems. References:

1. J. Medhi, Stochastic Processes, New Age International Publication, 2009. 2. S.M. Ross, Stochastic Process, John Wiley, 2008. 3. A. Papoulis and S.U. Pillai, Probability –Random Variables and Stochastic Processes, McGraw Hill Education,

4th edition, 2017. 4. S. Karlin and H.M. Taylor, A First Course in Stochastic Process, Academic Press, 2012. 5. E. Cinlar, Introduction to Stochastic Processes, Dover Books on Mathematics, 2013. 6. H.M. Taylor and S. Karlin, Stochastic Modeling, Academic Press, 1999.

MA7142: FUZZY SETS & THEIR APPLICATIONS [2 1 0 3]

Fuzzy Sets: Introduction, classical sets vs fuzzy sets, need for fuzzy sets, definition and mathematical representations,

level sets, fuzzy functions, Zadeh’s extension principle; Operations on Fuzzy Sets: Operations on [0, 1], fuzzy negation,

triangular norms, t-conorms, fuzzy implications, aggregation operations, fuzzy functional equations, fuzzy number;

Fuzzy Relations: Fuzzy binary and n-ary relations, composition of fuzzy relations, fuzzy equivalence relations, fuzzy

compatibility relations, fuzzy relational equations; Possibility Theory: Fuzzy measures, evidence theory, necessity and

belief measures, probability measures vs possibility measures; Approximate Reasoning: Fuzzy decision making, fuzzy

relational inference, positional rule of inference, efficiency of inference, hierarchical; Fuzzy Controllers: fuzzy if-then

rule base, inference engine, Takagi-Sugeno fuzzy systems, function approximation.

References:

1. A.K. Bhargava, Fuzzy Set Theory Fuzzy Logic and Their Applications, S. Chand & Co., 2013.

2. K. Pundir and R. Pundir, Fuzzy Sets and Their Applications, Pragati Prakashan, Meerut, 2008.

3. G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall of India Pvt. Ltd.,

New Delhi, 2001.

4. H. J. Zimmermann, Fuzzy Set Theory and its Applications, Springer, 2001.

DSE - II

MA7143: TOPOLOGY- II [2 1 0 3]

Separation Axioms: Regular, normal, T3 and T4 separation axioms, their characterization and basic properties,

Urysohn lemma and Tietze extension theorem, regularity and normality of a compact Hausdorff space, complete

regularity, complete normality, T5 spaces, their characterization and basic properties, product topological spaces,

projection mappings, Tychonoff product topology in terms of standard sub bases and its characterization, separation

axioms and product spaces, connectedness, locally connectedness and compactness of product spaces, product

space as first axiom space, Tychonoff product theorem; Embedding and Metrization : Embedding lemma and Tychonoff

embedding theorem, metrizable spaces, Urysohn metrization theorem; Nets : Nets in topological spaces, convergence

of nets, Hausdorffness and nets, subnet and cluster points, compactness and nets; Filters : Definition and examples,

collection of all filters on a set as a poset, methods of generating filters and finer filters, ultra filter and its

characterizations, ultra filter principle, image of filter under a function, limit point and limit of a filter, continuity in terms

of convergence of filters, Hausdorffness and filters, canonical way of converting nets to filters and vice versa, Stone-

Cech compactification, covering of a space, local finiteness, paracompact spaces, paracompactness as regular space,

Michaell theorem on characterization of paracompactness, paracompactness as normal space, A. H. Stone theorem,

Nagata- Smirnov metrization theorem.

References: 1. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, 2017. 2. K.D. Joshi, Introduction to General Topology, New Age International Private Limited, 2017 3. J. L. Kelly, General Topology, Springer Verlag, New York, 2000. 4. J. R. Munkres, Toplogy, Pearson Education Asia, 2002. 5. W.J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 2014. 6. K. Chandrasekhara Rao, Topology, Narosa Publishing House Delhi, 2009.

MA7144: LINEAR MODELS [2 1 0 3] Simple Regression Models: Straight line relationship between two variables, Gauss Markoff theorem, precision of the estimated regression, examination of the regression equation, lack of fit and pure error, fitting a straight line in matrix form, variance and covariance of b0 and b1 from the matrix calculation, variance of Y using the matrix development, orthogonal columns in the X-matrix, partial F-Test and sequential F-tests, selection of best regression equations by step wise procedure, bias in regression estimates, residuals, polynomial models and orthogonal polynomials; Multiple Regression Estimation: Introduction, the model, estimation of β and σ2, geometry of least- squares, the model in centered form, normal model, R2 in fixed x-regression, generalized least-square, model misspecification, tests of hypothesis and confidence intervals, model validation and diagnostics. References:

1. R.B. Bapat, Linear Algebra and Linear Models, 2nd edition Hindustan Book Agency, 1999. 2. C. R. Rao, Linear Models, Least Squares and Alternatives, 2nd edition, Springer, 1999. 3. D.C. Montgomery, E.A. Peck and G. G. Vining, Introduction to Linear Regression Analysis, 4 th edition, John

Wiley & Sons, 2006. 4. A. C. Rencher and G.B. Schaalje, Linear Models in Statistics, 2nd Edition, John Wiley & Sons, 2008. 5. S.R. Searle, Linear Models, Wiley Classic Library, Wiley-Inderscience, 1997.

MA7145: COMPUTATIONAL FLUID DYNAMICS [2 1 0 3] Introduction: Basic equations of Fluid dynamics, analytic aspects of partial differential equations classification, boundary conditions, maximum principles, boundary layer theory, finite difference and finite volume discretizations, vertex-centred discretization, cellcentred discretization, upwind discretization; Grid Analysis: Non uniform grids in one dimension, finite volume discretization of the stationary convection-diffusion equation in one dimension, schemes of positive types, defect correction, non-stationary convection diffusion equation; Stability of the system: Stability definitions, discrete maximum principle, incompressible Navier-Stokes equations, boundary conditions, spatial discretization on collocated and on staggered grids, temporal discretization on staggered grid and on collocated grid; Analytical Solutions: Iterative methods, stationary methods, Krylov subspace methods, multigrade methods, fast Poisson solvers, iterative methods for incompressible Navier-Stokes equations, Shallow-water equations – one and two dimensional cases, Godunov order barrier theorem, linear schemes, scalar conservation laws, Euler equation in one space dimension – analytic aspects, approximate Riemann solver of Roe, Osher scheme, flux splitting scheme, numerical stability, Jameson – Schmidt – Turkel scheme, higher order schemes. References:

1. P. Wesseling, Principles of Computational Fluid Dynamics, Springer Verlag, 2000. 2. J.F. Wendt, J.D. Anderson, G. Degrez and E. Dick, Computational Fluid Dynamics: An Introduction, Springer-

Verlag, 1996. 3. K. Muralidher, Computational Fluid Flow and Heat Transfer, Narosa Pub. House, 2013. 4. T.J. Chung, Computational Fluid Dynamics, Cambridge Uni. Press, 2014.