Unit I: Replacement Problems: Introduction, Failure mechanism of items, Replacement policy for items whose maintenance cost increases with time and money values is constant, Money value, Present worth factor and discount rate, Replacement policy for items whose maintenance cost increases with time and money values changes with constant rate, Individual replacement policy: Mortality theorem ,Group replacement policy.
Unit – II : Inventory – Introduction, Cost involved in inventory problems, variables in inventory problem, symbols in inventory , concept of Economic Ordering Quantity(EOQ), The EOQ models without shortage: Model I (a) The economic lot size system with uniform demand. Model I (b) Economic lot size with different rates of demand in different cycles. Model I (c) Economic lot size with finite Rate of Replenishment. (EOQ production model),EOQ model with shortages: Model II(a) The EOQ with constant rate of demand, scheduling time constant, Model II (c) The production lot size model with shortages. Probabilistic inventory Models, Instantaneous demand, no set up cost model Model VI(a) Discrete case Model VI(b) continuous case ,Problems on above models.
Unit – III : Queuing Theory - Queuing systems , Queuing Problems: transient and steady states,A list of symbols, traffic intensity, Probability distributions in Queuing systems: Distribution of arrivals ‘The Poisson process’, Properties of Poisson process of arrivals, Distribution of inter-arrival times(Exponential process).Distribution of Departures(Pure death model), Analogy of exponential service time with Poisson Arrivals Model I : ( M/M/I) : ( ∞ /FCFS):Birth and Death model ,Solution of model, Examples on model I, Problems.
Unit – IV: PERT / CPM : Applications of PERT /CPM techniques, Network diagram representation. Rules for constructing the Network diagram, Time estimates and critical path in network analysis, Examples on optimum duration and minimum duration cost, Problems
Recommended Books:
1. S.D.Sharma : Operations Research Kedarnath and co. 1999
Reference Books:
1. Hamdy Taha : Operations Research, Macmillan and Co.
2. J.K. Sharma : Operations Research, Macmillan India Ltd. 1999.
3. R.K.Gupta : Operations researcTime estimates h, Krishna Prakashan Mandir, 1999
- Teacher: Dayanand Gawade
Unit I: Review of divisibility: The division algorithm, G.C.D., Euclidean algorithm, Diophantine equation ax + by = c. Primes and their distribution : Fundamental theorem of Arithmetic, The Goldbach Conjecture.
Unit II : Congruences : Properties of Congruences, Linear congruences, Special divisibility tests. Fermat’s theorem : Fermat’s factorization method, Little theorem, Wilsons theorem. Number theoretic functions : The functions τ and σ. The Mobius Inversion formula, The greatest integer function.
Unit III: Euler’s Generalization of Fermat’s theorem: Euler's phi function, Euler's theorem, properties of phi function, An application to Cryptography. Primitive roots : The order of an integer modulo n.
Unit IV: Primitive roots for primes, composite numbers having primitive roots, The theory of Indices. The Quadratic reciprocity law: Eulerian criteria, the Legendre symbol and its properties, quadratic reciprocity, quadratic reciprocity with composite moduli .
Recommended Books : 1. D.M.Burton : Elementary Number Theory, Seventh Ed.MacGraw Hill Education(India)Edition 2012, Chennai. Reference Books : 1. S.B.Malik :Baisc Number theory, Vikas publishing House. 2. George E.Andrews : Number Theory, Hindusthan Pub. Corp.(1972). 3. Niven, Zuckerman : An Introduction to Theory of Numbers. John Wiley & Sons. 4. S. G. Telang , Number Theory, Tata Mc.Graw-Hill Publishing Co., New Delhi. 5. M.B. Nathanson, Methods in Number Theory, Springer(2009).
- Teacher: Dayanand Gawade
Unit I: Interpolation, differentiation and integration: Lagrange and Newton interpolations, Truncation error bounds, Newtons divided difference interpolation, finite difference operators, numerical differentiation, methods based on interpolation, numerical integration, methods based on interpolation, error analysis, Newton-Cotes methods, Error estimates for trapezoidal and Simpson’s rule.
Unit II: Numerical solution of differential equations: Euler method, analysis of Euler method, Backward Euler method, mid-point method, order of a method, Taylor series method, Explicit Runge-Kutta methods of order two and four, convergence and stability of numerical methods, Truncation error, error analysis.
Recommended Books:
1. M. K. Jain, S. R. K. Iyengar, R. K. Jain, Numerical methods for scientific and Engineering
Computation (Fifth Edition), New Age International Publishers 2007.
Reference Books:
1. S. S. Sastry, Introductory methods of Numerical Analysis (Fifth Edition), PHI learning Private
Limited, New Delhi 2012.
2. D. Kincaid, W. Cheney, Numerical Analysis Mathematics of Scientific Computing (Third
Edition), American Mathematical Society.
3. J.C. Butcher, Numerical methods for ordinary differential equations (Second Edition), John
Wiley & Sons Ltd, 2008.
4. Kendall E. Atkinson, An Introduction to Numerical Analysis (Second Edition), John Wiley &
Sons 1988.
- Teacher: Dayanand Gawade
Unit I : Sequences and series of functions: Pointwise convergence of sequences of functions, Examples of sequences of real valued functions, Definition of uniform convergence, Uniform convergence and continuity, Cauchy condition for uniform convergence, Uniform convergence and Riemann integration, Uniform convergence and differentiation
Unit II: Rearrangement of series, subseries, Double sequences, Double series, rearrangement of double series, sufficient condition for equality of iterated series, multiplication of series, Cesaro summability, sufficient conditions for uniform convergence of series, uniform convergence and double sequences, mean convergence, Taylor series generated by a function, Bernstein's theorem, binomial series.
Unit III: Multivariable differential Calculus: The Directional derivatives, directional derivatives and continuity, total derivative, total derivatives expressed in terms of partial derivatives, The matrix of linear function, mean value theorem for differentiable functions, A sufficient condition for differentiability, sufficient condition for equality of mixed partial derivatives, Taylor’s formula for functions from Rn to R1
Unit IV: Implicit functions: Functions of several variables, Linear transformations, Differentiation, Contraction principle, The inverse function theorem, The implicit function theorem and their applications.
Recommended books:
1. Mathematical Analysis, Apostal, Second Edition, Narosa Publishing House.1974
Reference books:
1.Principles of mathematical Analysis, Walter Rudin, third Edition, McGraw Hill book
company
2. Calculus Vol. I , Vol II, Tom M. Apostol, Second EditionWiley India Pvt. Ltd.
3. W.Fleming, Functions of several Variables,2nd Edition ,Springer Verlag, 1977.
- Teacher: Dayanand Gawade
Unit I : Sequences and series of functions: Pointwise convergence of sequences of functions, Examples of sequences of real valued functions, Definition of uniform convergence, Uniform convergence and continuity, Cauchy condition for uniform convergence, Uniform convergence and Riemann integration, Uniform convergence and differentiation
Unit II: Rearrangement of series, subseries, Double sequences, Double series, rearrangement of double series, sufficient condition for equality of iterated series, multiplication of series, Cesaro summability, sufficient conditions for uniform convergence of series, uniform convergence and double sequences, mean convergence, Taylor series generated by a function, Bernstein's theorem, binomial series.
Unit III: Multivariable differential Calculus: The Directional derivatives, directional derivatives and continuity, total derivative, total derivatives expressed in terms of partial derivatives, The matrix of linear function, mean value theorem for differentiable functions, A sufficient condition for differentiability, sufficient condition for equality of mixed partial derivatives, Taylor’s formula for functions from Rn to R1
Unit IV: Implicit functions: Functions of several variables, Linear transformations, Differentiation, Contraction principle, The inverse function theorem, The implicit function theorem and their applications.
Recommended books:
1. Mathematical Analysis, Apostal, Second Edition, Narosa Publishing House.1974
Reference books:
1.Principles of mathematical Analysis, Walter Rudin, third Edition, McGraw Hill book
company
2. Calculus Vol. I , Vol II, Tom M. Apostol, Second EditionWiley India Pvt. Ltd.
3. W.Fleming, Functions of several Variables,2nd Edition ,Springer Verlag, 1977.
- Teacher: Dayanand Gawade
Unit I: Topological Spaces, Basis and Subbasis for a Topology, The Order Topology, The Product Topology on 𝑋×𝑌, The Subspace Topology.
Unit II: Closed Sets , Closure and Interior of a Set, Limit Points, Hausdorff Spaces, Continuity of Functions, Homeomorphisms, The Product Topology, The Metric Topology.
Unit III: Connected Spaces, Connected Subspaces of the Real Line, Components and Local Connectedness, Compact Spaces, Compact Subspaces of the Real Line.
Unit IV: The Countability Axioms, The Separation Axioms, Normal Spaces, The Urysohn Lemma, The Urysohn Metrization Theorem (Only statement and its importance), The Tietze Extension Theorem (Only statement and its importance).
Recommended Book:
1. J. R. Munkers, Topology, Second Edition, Pearson Education (Singapore), 2000.
Reference Books:
1. W. J. Pervin, Foundations of General Topology, Academic Press, New York, 1964.
2. J. L. Kelley, General Topology, Springer-Verlag, New York, 1955.
3. S. Willard, General Topology, Addison-Wesley Publishing Company, 1970. 4. K. D. Joshi, Introduction to General Topology, New Age International, 1983.
G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw Hill Book
Company, New Delhi, 1963.
- Teacher: Dayanand Gawade
Unit I: Group of permutations, Examples, Alternating Groups, Simple groups, simplicity of An (n > 4), Applications, Subnormal and Normal Series, Jordan-Holder Theorem, The Center and the Ascending Central Series, Isomorphism Theorems.
Unit II: The Zassenhaus (Butterfly) Lemma, Schreier Theorem, Group action on a set, fixed sets and isotropy subgroups, Orbits, Applications of G-Sets to Counting, Burnside theorem, p-groups, The Sylow Theorems.
Unit III: Applications of Sylow theorems to p-Groups and the Class equation, Further Applications, Polynomial in an Indeterminate, Polynomial rings, The evaluation Homomorphisms, Factorization of Polynomials over Fields, The Division Algorithm in F[x], Irreducible Polynomials, Eisenstein criteria, Ideal Structure in F[x], Uniqueness of Factorization in F[x].
Unit IV: Principal Ideal Domain (PID), Uniqueness of Factorization Domain(UFD), Gauss lemma, Introduction and Definition of Euclidean Domain, Arithmetic in Euclidean Domains. Definitions and Examples of Modules, Direct Sums, Free Modules, sub-modules, Quotient Modules, Homomorphism, Simple Modules
- Teacher: Dayanand Gawade
