B.Sc. (Hons) Maths

Semester VI

  • COURSE CODE
    COURSE NAME
    CREDITS
  • JBMH-601

    Analysis-III:

    UNIT-I

    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.

    UNIT-II

    Fundamental theorems of Calculus. (Second fundamental theorem without proof),Consequences of the Fundamental theorems of Calculus integration of parts and change of variables, mean value of calculas (Statement only), improper integrals, convergences of improper integrals, abel’s and dirichlet’s test for improper integrals, Beta & gamma functions and their relations.

    UNIT-III

    Pointwise & uniform convergence of sequence of function uniform convergence & continuity, uniform convergence & differentiation, uniform convergence &integration, Cauchy criterian for uniform convergence, series of function and convergence, weirstrass M- test, weirstrass approximation theorem ( Statement only)

    UNIT-IV

    Fourier series, Bessel’s inequality, Dirichlet’s criteria of convergence of fourier series, fourier series for even & odd function, fourier series on [o, 2π] change of intervals, Half range sine & cosine series.

    Books:

    • 1. K.A. Ross, Elementary Analysis: The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.
    • 2. R.G. Bartle D.R. Sherbert, Introduction to Real Analysis (3rd edition), John Wiley and Sons (Asia) Pvt. Ltd.., Singapore, 2002.
    • 3. Charles G. Denlinger, Elements of Real Analysis, Jones and Bartlett (Student Edition).
    04
  • JBMH-602

    Algebra-III:

    UNIT-I

    Definition and examples of homomorphism, properties of homomorphism, definition and examples of isomorphism, Cayley's Theorem, properties of isomorphism, Isomorphism theorems I, II, III.

    UNIT-II

    Definitions and examples of automorphism, inner automorphisms, automorphism and inner automorphism group, automorphism group of finite and infinite cylic groups, applications of factor groups to automorphisms groups, Cauchy theorem for finite abelian groups.

    UNIT-III

    Definition and examples of rings, properties of rings, subrings, integral domains and fields, characteristic of a ring, Ideals, ideal generated by a subset of a ring, factor rings, operations on ideals, prime and maximal ideals.

    UNIT-IV

    Ring homomorphism, properties of ring homomorphism, Isomorphism theorems I, II and III, field of quotients.

    • 1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Edition), NarosaPublishing House, New Delhi, 1999.
    04
  • JBMH-603

    Operation Research:

    UNIT-I

    Introduction to linear programming problem, Formulation of LPP, Graphical method, Theory of simplex method, optimality and unboundedness, the simplex algorithm, introduction to artificial variables, two phase method, Big-M method.

    UNIT-II

    Duality, Formulation of dual problem, primal dual relationships, economic interpretation of the dual.

    UNIT-III

    Transportation problem and its mathematical formulation, northwest-corner method, least cost method and Vogel approximation method for determination of starting basic solution, algorithm for solving transportation problem, assignment problem and its mathematical formulation, Hungarian method for solving assignment problem.

    UNIT-IV

    Game theory: formulation of two person zero sum games, solving two person zero sum games, games with mixed strategies, graphical procedure, linear programming solution of games.

    • 1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear programming and network Flows, John Wiley and Sons, India.
    • 2. F.S. Hillier and G.J. Liberman, Introduction to operation research (9th edition), Tata McGrew Hill, Singapore, 2009.
    • 3. Operation Research, S.D. Sharma, Kedar Nath Ram Nath, Meerut, Delhi.
    • 4. Introduction to Operation Research by Fredrick S.Hillier, Genrald J. Liberman, Bodhibrata Nag, Preetam Basu, Tata Mc Grew Hill Education,Pvt. Ltd.
    • 5. Operation Research by Prof. Vivek Kumar, S.K. Kataria and Sons.
    • 6. Operation Research by G. Srinivasan, PHI Learning, Pvt Ltd, New Delhi-110001.
    04
  • JBMH-604

    Elective Paper III (Hydrodynamics/ Probability Theory)

    Hydrodynamics

    UNIT-I

    Equation of continuity ,Equation of continuity in cartesion, cylindrical and polar form, acceleration of a fluid particle, boundary surfaces, Lagranges & Euler methods elation between , Lagranges & Euler methods.

    UNIT-II

    Equation of motion, , Lagranges & Euler’s equation of motion, conservative field of force, equation of motion under impulsive force, Bournoli’s Equation, Cauchy’s equation.

    UNIT-III

    Motion in Two dimensions, stream function and current function, complex potential function, source, sink, doublet, image in two and three dimensions, Circle theorem and Blasius theorem.

    UNIT-IV

    Motion of sphere through infinite mass of liquid, liquid streaming pass a fixed sphere, Stoke stream function.

    • 1. "Fluid Mechanics" by A.S. Ramsay and W. H. Besant, C.B.S. Publisher’s Pvt. Ltd. Agra
    • 2. "Fluid Dynamics" by F. Chroltan, E.L.B.S. Van Nostrand Co.
    • 3. "Fluid Dynamics" by J. K. Goyal and K. P. Gupta, Pragati Prakashan
    • 4. "Fluid Dynamics" by B.D. Gupta, Pragati Prakashan
    • 5. "Fluid Dynamics" by M.D. Rai Singhania , S. Chand and Co.
    • 6. "Fluid Dynamics" by P.P Gupta, S. Chand and Co.

    Probability Theory:

    UNIT-I

    Random experiment, sample space, axioms of probability, elementary properties of probability, equally likely outcome problems. Random variables: Concept, cumulative distribution function, discrete and continuous random variables, expectation, mean, variance, moment generating function.

    UNIT-II

    Discrete random variables: Bernoulli random variable, Binomial random variable, geometric random variable, Poisson random variable. Continuous random variables: Uniform random variable, exponential random variable, Gamma random variable, Normal random variable.

    UNIT-III

    Conditional probability and conditional expectations, Bayes theorem, independence, computing expectation by conditioning; some applications — a list model, a random graph, Polya's urn model.

    UNIT-IV

    Bivariate random variables: Joint distribution, joint and conditional distributions, the correlation coefficient. Bivariate normal distribution.
    Functions of random variables: Sum of random variables, the laws of large numbers, central limit theorem, approximation of distributions.
    04
  • Total Credits
     
    16