Mathematics Syllabus For M .Sc OU

I am M sc first semester Mathematics student from Osmania University I have not my syllabus so can you provide me procedure how I can get m sc first semester syllabus?

Following is M.Sc. course syllabus of Osmania University:

Semester - 1
Software Project Management
Advanced Operating Systems
Soft Skills
Algorithms
Artificial Intelligence
Finanancial and Managerial Accountancy

Semester-2
Customer relation Management
Knowledge Management
Functional Management Elective
Supply Chain Management and Logistics
Data WareHousing and Data Mining
Soft Skills
Software Reuse Techniques
Network Security
Information Systems Audit and Control


Semester - 3
Enterprise Application Integration
Is policy and Strategy
Specialization Electives (yet to be decided)
Soft Skills

Semester - 4
Specialization Electives
Project

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Osmania University was established in the year 1918, it offers M .Sc Mathematics through the Department Of Mathematics & Statistics. This is the syllabus of program:

M. sc. mathematics syllabus:
I Semester - (common for Mathematics, Applied Mathematics & Statistics-OR)
MM 500 Real Analysis
MM 501 Linear Algebra
MM 502 Discrete Mathematical Structures
MM 503 C Programming
ST 500 Elements of Probability & Statistics
ST 501 Linear Programming
MM 500 REAL ANALYSIS-I
Real number system and its structure, Infimum, Supremum, Dedekind cuts. (proofs omitted)
Sequences and Series of real numbers, Subsequences, Monotone sequences, Limit infimum, Limit
Supremum, Convergence of Sequences and Series, Cauchy criterion, Root and Ratio tests for the
convergence of series, Power series, Product of Series, Absolute and Conditional convergence. Metric
spaces, limits in metric spaces.
Functions of single real variable, Limits of functions, Continuity of functions, Uniform continuity,
Continuity & compactness, Continuity and connectedness, Types of discontinuities, Monotonic
functions, Infinite limit and Limit at infinity.
Differentiation, Properties of derivatives, Chain rule, Rolle's theorem, Mean-value theorems,
L'Hospital's rule, Derivatives of higher order, Taylor's theorem.
Sequences and Series of functions, Pointwise and uniform convergence, Uniform convergence of
continuous functions, Uniform convergence and differentiability, Equicontinuity, Pointwise and
uniform boundedness, Ascoli's theorem, Weierstrass approximation theorem, Fourier series.
REFERENCES:
1. R.R.Goldberg, Methods of Real Analysis.
2. W. Rudin, Principles of Mathematical Analysis (Units 1-5 except Fourier series).
3. R.G. Bartle, The Elements of Real Analysis (only for Fourier Series), 2nd Ed., J .Wiley, NY, London.
4. Kenneth A. Ross, Elements of Analysis: The Theory of Calculus, Springer Verlag, UTM, 1980.
MM 501 LINEAR ALGEBRA
Matrices: Elementary operations, reduced Row-Echelon form; consistency of system of equations,
solutions of systems of equations, homogeneous system, inverse of a Matrix, Determinants, Cramer's
Rule.
Vectors, Inner Product, C-S inequality, Metric in R , triangular inequality, Vector spaces and subspaces,
Linear independence of vectors, Basis, Orthonormal basis, Gram-Schmidt construction of orthonormal
basis.
Linear transformations and matrices, kernel, Nullity theorem, Rank of a matrix, Similarity.
Characteristic polynomials, Eigen values, Theorems on Eigen values and Eigen vectors. Cayley-
Hamiltonian theorem. Properties of characteristic polynomials, direct sum, Jordan form and
diagonalization. Page 3

Left and right inverse of matrices, g-inverse, Algorithm for evaluation of Moore-Penrose g-inverse of a
matrix, proof of its uniqueness, bilinear and quadratic forms, their properties.
REFERENCES:
1. Hoffman and Kunze, Linear Algebra.
2. Rao A.R., Bhimashankaram P., Linear Algebra. (Tata Mc-Graw Hill)
MM 502 DISCRETE MATHEMATICAL STRUCTURES
* Relations and Functions: Introductions, Properties of Binary Relations, Equivalence Relation and
Partitions, Partial Ordered Relations and Lattices, Chains and Antichains, Functions and the Pigconhole
Principle.
* Discrete Numeric Functions and Generating Functions: Introduction, Manipulation of Numeric
Functions, Asymptotic Behaviour of Numeric Functions, Generating functions, Combinatorial
Problems.
* Recurrence Relations and Recursive Algorithms: Introduction, Recurrence Relations, Linear
Recurrence Relations with Constant Coefficients, Homogeneous Solutions, Particular Solution, Total
Solutions, Solution by the Method of Generating Functions, Introduction to Algorithms, Sorting
Algorithms, Matrix Multiplication Algorithm.
* Groups and Rings: Introduction, Semi-groups, Groups, Subgroup, Generators and Evaluation of
Powers, Cosets and Lagrange's Theorem, Permutation Groups Cayley’s Theorem, Burnside's Theorem,
Sylow’s Theorem, Codes and Group Codes, Isomorphisms and Autormorphisms, Homomorphisms
and Normal subgroxups, Coding Methods based on Entropy (Shannon-Fano, Huffman Codes), Discrete
Source and the First Coding Theorem.
* Boolean Algebras: Lattices and Algebraic Systems, Principle of Duality, Bsaic Properties of
Algebraic Systems, Distributive and Complemented Lattices, Boolean Lattices and Boolean Algebras,
Uniqueness of Finite Boolean Algebras, Boolean Functions and Boolean Expressions, Normal Forms
of Boolean Expressions and Simplications of Boolean Expressions, Prepositional Calculus, Design
and Implemented of Digital Neworks Switching Circuits.
Books:
* Liu "Elements of Discrete Mathematics" McGraw Hill
* Tremblay and Manohar "Discrete Mathematics Structures with Applications to Computer Science"
(1997) McGraw Hill
* Black "Applied Probability" John Willy
MM503 “C” PROGRAMMING
C language : Programmer's model of a computer, Algorithms, Flow Charts; Data Types, Arithmetic and
input/output instructions; Decisions control structures; Decision statements, Logical and Conditional
operators; Loop, Case control structures; Functions; Preprocesssors; Arrays; Puppettting of strings;
Structures; Pointers; File formatting.
Algorithms and programms to analyze statistical data and solve routine statistical problems. Measures
of location and dispersion, sorting of data, solving systems of equations constructing inverse matrices and
g-inverses.
Numerical Algorithms using C.
REFERENCES: Page 4

1. Henry Mullish & Hobert Looper, Spirit of C: An Introduction to Modern Programming,
Jaico Publishers, Bombay.
2. Kernighan B.W. and Ritchie D.M., C Programming Language, Prentice Hall, Software Series.
ST 500 ELEMENTS OF PROBABILITY AND STATISTICS
Random experiments, Sample spaces, Sets, Events, Algebras. Elements of combinatorial analysis.
Classical definition and calculation of Probability, Independence of events.
Random variables; Distribution functions, Moments, Probability and Moment generating functions,
Independence of random variables.
Introduction to various discrete and continous random variables, Limiting distributuions of some
random variables, Distributions of functions of random variables. Bi-variate distributions, Conditional and
marginal distributions, Conditional expectation & variance. Co-varaiance and correlation co-efficient.
Elemetary understanding of data : Frequency curves, Emparical measures of location, spread; Empirical
moments, Analysis of bivariate data; Fitting of distributions.
REFERENCES:
Chakravarthy, I.M.
: Handbook of Applied Statistics (Willey).
Feller, W.
: Introduction to Probability Theory and its Applications (Willey - Eastern).
Mood, A.M., Graybill, F. : Introduction to the Theory of Statistics (Mc-Grawhill).
Gnedenko, B.
: The Theory of Probability (MIR, Moscow).
ST 501 LINEAR PROGRAMMING
Graphical solutions in 2-dimensions, simplex method, duality theorem, relation to conservative 2 -
person games. Parametric linear programming, transportation and assignment problems, PERT & CPM.
REFERENCES:
Hadley, G.,
: Linear Programming (Addison - Wesley)
Kambo, N.S.,
: Mathematical Programming Page 5

II Semester – Mathematics, Applied Mathematics
MM 421 Real Analysis-II
MM 422 Complex Complex
MM 423 Measure & Integration
MM 424 Topology
MM 541 Algebra-II
MM 421 REAL ANALYSIS-II
Unit-1:
Functions of several-variables, Directional derivative, Partial derivative, Total derivative,
Jacobian, Chain rule and Mean-value theorems, Interchange of the order of differentiation, Higher
derivatives, Taylor's theorem, Inverse mapping theorem, Implicit function theorem, Extremum
problems, Extremum problems with constraints, Lagrange's multiplier method.
Unit-2:
Multiple integrals, Properties of integrals, Existence of integrals, iterated integrals, change of
variables.
Unit-3:
Curl, Gradient, div, Laplacian cylindrical and spherical coordinate, line integrals, surface integrals,
Theorem of Green, Gauss and Stokes.
REFERENCES:
1. Apostol T.M., Mathematical Analysis (Original Edition) Ch. 6,7,10 & 11
2. Apostol T.M., Calculus-II - Part-2 (Non-Linear Analysis)
3. Vector Analysis (Schaum Series)
MM 422 COMPLEX VARIABLES - I
Algebra of complex numbers; Operations of absolute value and Conjugate; Standard inequalities for
absolute value, extended complex plane, spherical representation, and neighborhoods of ì and (C) as a
metric space and its topological properties.
Concept of analytic function via power series and differentiability methods. The exponential and
logarithmic functions, trigonometric functions of a complex variable. Analytic functions as mappings
from C to C. Conformality of a map linear fractional transformations and their properties and
elementary conformal mappings. Examples.
Complex Integration: Line integrals, rectifiable curves; Cauchy's fundamental theorem for rectangle, disk,
index of a closed curve, Cauchy's integral formula, higher derivatives of analytic functions,
Cauchy's inequality, Liouville's theorem.
Singularities: Taylor's theorem, removable singularities, zeros and poles, local mapping essential
singularities, examples, Weierstrass theorem, the maximum modulus theorem, Schwartz's lemma,
Cauchy's residue theorem; evaluation of definite integrals using Cauchy's residue theorem; Argument
principles; Taylor Series and Laurent series, expansions, Examples.
REFERENCES:
1. Ahlfors, Complex Analysis Page 6

2. Churchill, Brown, Complex Analysis - Ed. V
3. Conway, Functions of One Complex Variable
MM 423 MEASURE AND INTEGRATION
Riemann-Stieltjes Integral, Riemann's condition, Linear properties of Integration, Necessary
conditions for existence of Riemann- Stieltjes Integrals, Sufficient conditions for existence of
Riemann-Stieltjes Integrals, Reduction to Riemann Integral, Change of Variable in a Riemann-Stieltjes
Integral, Comparison Theorems, Mean-value theorems for Riemann-Stieltjes Integrals, Integral as a
function of the interval, Fundamental theorem of Integral calculus, Improper integrals and tests for
their convergence, Absolute convergence.
σ-algebras of Sets, Borel subsets of R, Lebesgue outer measure and its properties, σ-algebra of
measurable sets in R, Non-measurable set, Example of measurable set which is not a Borel Set, Lebesgue
Measure and its properties, Lebesgue-Stieltjes measure, Measurable function pointwise convergence
and convergence in measure, Egoroff theorem, Lebesgue integral, Lebesgue criterion of Riemann
integrability, Fatou's Lemma, convergence theorems, Differentiation of an integral, Absolute continuity
with respect to Lebesgue measure.
Lebesgue Integral in the Plane, Introduction to Fubini's theorem. Lp-spaces.
REFERENCES:
1. Apostol T.M., Mathematical Analysis (Chapter-7)
2. Bartle R.B., Elements of Real Analysis (Chapter 5, Sec. 32)
3. H.L.Royden, Real Analysis (Chapter 3-6)
MM 424 TOPOLOGY
Definition of Topologies in terms of open sets, neighborhood system, closed sets and closure operations
and their equivalence, points of accumulation, interior, exterior and boundary points. Base and sub-
base of a topology, subspace, product space, quotient space, continuous, open and closed maps,
homeomorphism convergence of sequence and filters, separation axioms, separability, Lindeloff
space, Urysohn's metrization theorem, compactness, local compactness, sequential and countable
compactness, Tychonov theorem, one point compactification, connectedness and local connectedness.
REFERENCES:
1. Dugundji J., Topology.
2. Munkres, Topology.
MM 541 ALGEBRA-II / RINGS AND MODULES
1. Basic concepts in Rings, Ideals, Homomorphism of rings, quotients with several examples.
2. Euclidean domains, principal ideal & unique factorization domains.
3. Eisensteins irreducibility criterion and Gauss's lemma.
4. Basic concepts in Modules, submodels, Homomorphisms, quotients, direct sum.
5. Simple Modules, Cyclic modules and Modules over p.i.d's.
6. Modules with chain condition.
REFERENCES:
1. Musili C., Introduction to Rings and Modules, (Narosa, 1992)
2. Jacobson N., Basic Algebra
3. Artin M., Algebra Page 7

II Semester (Statistics)
Probability & Measure Theory
Theory of Sampling
Linear Models
Statistical Methods
Theory of Inference-I
PROBABILITY AND MEASURE THEORY
Probability on Boolean algebras, extension of probability measure, random variables and vectors.
Integration : DCT, LDCT, expectation. Modes of convergence of random variables, Laws of large
numbers. Characteristic function : Properties, Inversion theorem and Characterization. Central Limit
Theorems and corollaries.
REFERENCES:
Billingsley, P.,
Probability and Measure (John - Willey)
Parthasarathy, K.R.,
Introduction to Probability and Measure (Tata - McGrawhill)
THEORY OF SAMPLING
Questionnaire, scale construction, item analysis, reliability and validity of scores.
Organizations of sample surveys, simple random sampling with and without replacement, stratified
random sampling, systematic sampling, cluster and multistage sampling, varying probability
sampling,interpenetrating subsamples.
Ratio and regression methods of estimation.
Control of non-sampling errors and non-response.
TEXT:
Cochran, W.G.
:
Sampling Techniques (John Wiley)
Des Raj
:
Sampling Theory (Tata Mc Graw Hill)
Murthy, M.N.
:
Sampling Theory and Methods
Sukhatame, P.V.
:
Sampling Theory of Surveys with Applications
Sukhatame, B.V.
(Indian Soc. of Agri. Stats.)
Sukhatame, S. & Ashok, C.
LINEAR MODELS
Linear estimation and theory of least squares, Gauss - Markov theorem: BLUE for linear parametric
functions, Tests of linear hypothesis. ANOVA for one way and two way classifications, variance
component model. Regression analaysis, Step wise regression, Dummy variables, Analysis of
covariance, Polynomial regression and Orthogonal; regression.
Basic princoiples of experimental design: Randomization, Structure and analysis of completely
randomized, randomized blocks and Latin square designs. Factorial experiments: analysis of 2 and 3
factorial experiments in randomized blocks.
REFERENCES: Page 8

Kshirsagar, A.N.
:
Linear Models and Applications (Marcel - Decker)
Rao, C.R.
:
Linear Statistical Inference and Applications (Willey - Eastern)
Montgomery, D.C.
:
Design and Analysis of Experiments (John - Willey)
STATISTICAL METHODS
Introduction to the concepts of statistical inference with examples from discrete distributions.
Estimation and tests for mean and variance for normal distribution - one & two population cases.
Correlation coefficient - evaluation and tests; simple linear regression, comparision of k-linear
regressions, fitting polynomial regressions & orhtogonal polynomials and related tests.
Explanatory data Analysis and robust techniques. Analysis of discrete data. Introduction to interval
estimation.
Some Non-Parametric tests - sign, run, median, Mann - Whitney – Wilcoxon. Spearman and Kendall ç
tests.
Numerical Methods used in Statistics.
REFERENCES:
Snedcor, G.W., Cochran, W.G. : Statistical Methods (Oxford & IBH).
Conover, W.J.,
: Practical Non-Parametric Statistics (JW)
THEORY OF INFERENCE - I
Methods of estimation, properties of estimators - unbaisedness, consistency, sufficiency,
efficciency, completeness. MVUE ; Rao - Blackwell theorem, Lehmann-Scheffe theorem, Crammer -
Rao inequality, Chapman - Robins inequality.
Interval estimation. Hypothesis testing: simple and composite hypotheses, Types of errors,
Power function, Neymann - Pearson lemma, Most powerful tests.
REFERENCES:
Kendal, M.G. & Stuart, A.
:
Advanced Theory of Statistics, Vol.1 (Charles - Griffin)
Lehmann, E.L.
:
Theory of Point Estimation (Wiley-Eastern)
Lehmann, E.L.,
:
Testing of Hypothesis (Willey - Eastren) Page 9

III Semester (Mathematics, Applied Mathematics)
Mathematics:
Applied Mathematics:
MM 551Ordinary Differential Equations
MM 551 Ordinary Differential Equations
MM 552 Functional Analysis
MM 552 Functional Analysis
MM 553 Partial Differential Equations
MM 553 Partial Differential Equations
MM 555 Mathematical Methods
MM 555 Mathematical Methods
MM 554 Galois Theory
MC 551 Numerical Analysis
MM 551 ORDINARY DIFFERENTIAL EQUATIONS
Ordinary Differential Equations, Mathematical Models, First Order Equations, Existence - Uniqueness
and continuity theorems, separation and comparison theorems, system of equations existence theorems,
Homogeneous linear systems, Nonhomogeneous Linear systems, Linear systems with constant
coefficients.
Two-point boundary-value problem, Green's functions, Construction of Green's functions,
Nonhomogeneous boundary conditions, Sturm-Liouville Systems, Eigen values and Eigen functions,
Eigen function expan- sions convergence in the Mean, Autonomous systems, Stability for Linear
systems with constant coefficients, Linear plane autonomous systems, perturbed systems, Method of
Lyapunov for nonlinear systems. Limit cycles of Poincare.
REFERENCES:
Simmons
: Ordinary Differential Equations
MM 552 FUNCTIONAL ANALYSIS
Fundamentals of normed linear spaces, Hahn-Banach theorem bounded linear maps on banach spaces,
open mapping and closed graph theorems, uniform boundedness principle, Duals and transpose, Duals of
Lp(I≤p<∞), C[a,b]. Definition of reflexivity, and examples.
Inner product spaces, Hilbert spaces, Projection Theorem, Riesz representation theorems. Contraction
mapping theorem and applications to differential and integral equations.
REFERENCES:
1. Kreyzig, Elements of Functional Analysis
2. Limaye B.V., Functional Analysis
MM 553 PARTIAL DIFFERENTIAL EQUATIONS
FIRST ORDER P D E:
Surfaces and Curves-Classification of Ist order p.d.e. Classification of solutions-Pfaffian differential
equations - Quasi-linear equations, Lagrange's method-compatible systems-Charpit's method- Jacobi's
method-Integral surfaces passing through a given curve- method of characteristics for quasi-linear and
non-linear p.d.e., Monge cone, characteristic strip. Page 10

SECOND ORDER P D E:
Origin of second order p.d.e's - classification of second order p.d.e's. Wave equation - D'Alemberts'
solution - vibrations of a finite string - existence and uniqueness of solution - Riemann method. Laplace
equation - boundary value problems - Maximum and minimum principles - Uniqueness and continuity
theorems - Dirichilet problem for a circle - Dirichilet problem for a circular annulus - Neumann problem
for a circle - Theory of Green's function for Laplace equation. Heat equation - Heat conduction
problem for an infinite rod - Heat conduction in a finite rod - existence and uniqueness of the
solution. Classification in higher dimensions - Kelvin's inversion theorem - Equipotential surfaces.
REFERENCES:
1. Tyn Myn T.U., Partial Differential Equations (Chapters 2,4,6,8)
2. John F., Partial Differential Eqautions (II Edn.) (Springer Verlag) Chapter - I
3. Weinberger H.F., Intro. to Partial Diff.Equations
4. Ian Sneddon, Elements of Partial Diff.Equations (Chapters 1,2,4)
5. Greenspan, Intro. to Partial Diff. Equations
6. Copson E.T., Classical Analysis
MM 555 MATHEMATICAL METHODS
INTEGRAL TRANSFORMS:
Laplace transforms: Definitions - properties - Laplace transforms of some elementary functions -
Convolution Theorem - Inverse Laplace transformation - Applications.
Fourier transforms - Definitions - Properties - Fourier transforms of some elementary functions -
Convolution theorems - Fourier transform as a limit of Fourier Series - Applications to PDE.
INTEGRAL EQUATIONS:
Volterra Integral Equations: Basic concepts - Relationship between Linear differential equations and
Volterra integral equations - Resolvent Kernel of Volterra Integral equation - Solution of Integral
equations by Resolvent Kernel - The Method of successive approximations - Convolution type equations,
solution of integral differential equations with the aid of Laplace transformation.
Fredholm Integral equations: Fredholm equations of the second kind, Fundamentals - Iterated
Kernels, Constructing the resolvent Kernel with the aid of iterated Kernels - Integral equations with
degenerate Kernels - Characteristic numbers and eigen functions, solution of homogeneous integral
equations with degenerate Kernel - nonhomogeneous symmetric equations - Fredholm alternative.
CALCULUS OF VARIATIONS:
Extrema of Functionals: The variation of a functional and its properties - Euler's equation - Field of
extremals - sufficient conditions for the Extremum of a Functional conditional Extremum Moving
boundary problems - Dis continuous problems - one sided variations - Ritz method.
REFERENCES:
1. Sneddon I., The Use of Integral Transforms (Tata McGraw Hill)
2. Schaum's Series, Laplace Transforms
3. Gelfand and Fomin, Calculus of Variations (Prentice Hall, Inc.) Page 11

4. Krasnov, Problems and Exercises in Calculus of Variations (Mir Publ.)
5. Ram P Kanwal, Linear Integral Equations (Academic Press)
MM 554 GALOIS THEORY
Field theory and Compass constructions: Algebraic, Complex algebraic numbers, Number fields;
transcendental, separable, normal purely inseparable extensions; finite fields; the Frobenius of a
field of positive characteristic; Perfect fields; theorem of the primitive element; Ruler and Compass
constructions; constructing regular polygons;
Galois theory and applications: Group of automorphisms of fields; fundamental theorem of finite Galois
Theory; cyclic extensions; solvability by radicals; Kummer theory; Determining the Galois group of a
polynomial;
Transcendental extensions: Transcendence basis theorem; Luroth's theorem; ranscendence of e.
Algebraically closed fields: Existence and uniqueness of an algbebraic closure.
REFERENCES:
1. Garling D.J.H., Galois Theory (Cambridge Univ. Press)
2. Stewart I.N., Galois Theory (Chapman Publ. Co.)
3. Jacobson N., Basic Algebra (Hindustan Publ. Corpn., Delhi) Vol. 1 (Chap. 4), Vol. 2 (Chap. 8)
4. Jacobson N., Lectures on Abstract Algebra Vol.3
5. Lang S., Algebra (Adison Wiley)
MC 551 NUMERICAL ANALYSIS
Numerical Computation:
Representation of integers and fractions, fixed point and floating point arithmetics, error propagation,
loss of significance, condition and instability, computational method of error propagation.
Polynomial Interpolation:
Existence and uniqueness of interpolation polynomial, Interpolation using differences, error of the
interpolating polynomial, osculatory interpolation, piecewise-polynomial approximation.
Solution of Nonlinear Equations:
Iterative methods, fixed point iteration, convergence of methods, polynomial equations, Miller's method,
convergence acceleration.
Solution of Linear Systems:
Elimination with and without pivoting, triangular factorization, error and residual of an approximate
solution. Backward errors and iterative improvement, fixed point iteration and relaxation methods.
Numerical Integration:
Basic rules of numerical integration, Gaussian rules, composite rules, adaptive quadrature,
Extrapolation to the limit, Romberg Integration.
Solution of ODEs:
Numerical differentiation, difference equations, Taylor series method, Euler's method and its
convergence, Runge-Kutta methods, Multistep formulas, Predicator-Corrector methods, Adams-
Moulton method, Stability of numerical methods, Round-off error propagation & control, shooting
methods and finite difference methods for BVPs.
REFERENCES:
1. Conte S.D. and deBoor C., Elementary Numerical Analysis - An Algorithmic Approach; 3rd edn., Page 12

McGraw Hill, 1981.
2. Henrici P., Elements of Numerical Analysis (John Wiley & Sons, 1964).
3. Froberg C.E., Numerical Mathematics - Theory and Computer Applications;
The Benjamin Cummings Pub. Co. 1985. Page 13

Rest of the syllabus is in the given below word file please have a look on that

Contact details:
Osmania University
Osmania University Main Rd, हैदराबाद, Andhra Pradesh 500007 ‎
040 2709 8043

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