MSc Mathematics syll III IV CBCS 1 Syllabus Mumbai University

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UNIVERSITY OF MUMBAI
Syllabus
for
M.A./M.Sc. Semester III & IV (CBCS)
Program: M.A/M.Sc.
Course: Mathematics
with eect from the academic year 2018-2019
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M.A./M.Sc. Semester III and IV
Choice Based Credit System (CBCS)
Semester III
Algebra III
Course Code Unit Topics Credits L/W
PSMT301,PAMT301Unit I Groups
Unit II Representation of nite groups 6 4
Unit III Modules
Unit IV Modules over PID
Functional Analysis
Course Code Unit Topics Credits L/W
PSMT302,PAMT302Unit I Baire spaces, Hilbert spaces Rn
Unit II Normed linear spaces 6 4
Unit III Bounded linear maps
Unit IV Basic theorems
Dierential Geometry
Course Code Unit Topics Credits L/W
PSMT303,PAMT303Unit I Isometries of Rn
Unit II Curves 6 4
Unit III Regular surfaces
Unit IV Curvature
Elective Courses
PSMT304,PAMT304 Elective Course I 3 4
PSMT305,PAMT305 Elective Course II 3 4
Note:
1. PSMT301/PAMT301, PSMT302/PAMT302, PSMT303/PAMT303 are compulsory courses
for Semester III.
2. PSMT 304/PAMT 304 and PSMT 305/PAMT305 are Elective Courses for Semester III.
3. Elective course Courses I and II will be any TWO of the following list of ten courses:
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1 ALGEBRAIC TOPOLOGY
2 ADVANCED COMPLEX ANALYSIS
3 COMMUTATIVE ALGEBRA
4 ALGEBRAIC NUMBER THEORY
5 PARTIAL DIFFERENTIAL EQUATIONS
6 NUMERICAL ANALYSIS
7 GRAPH THEORY
8 DESIGN THEORY
9 CODING THEORY
10 INTEGRAL TRANSFORM
4. In Semester III, there is a Skill course which is mandatory. Separate fees shall be collected
for the Skill course, the quantum of which shall depend on the nature of the skill course.
Teaching Pattern for Semester III
1. Four lectures per week for each of the courses: PSMT301/PAMT301, PSMT302/PAMT302,
PSMT303/PAMT303, PSMT304/PAMT304 and PSMT305/PAMT305. Each lecture is
of 60 minutes duration.
2. In addition, there shall be tutorials, seminars as necessary for each of the ve courses.
3. The lectures of the Skill Course are held on Sundays or other Holidays. This course shall
be approximately 100 hours duration. 75% attendance is mandatory for this course.
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Semester IV
Field Theory
Course Code Unit Topics Credits L/W
PSMT401,PAMT401Unit I Algebraic Extensions
Unit II Normal and Separable Extensions 6 4
Unit III Galois theorems
Unit IV Applications
Fourier Analysis
Course Code Unit Topics Credits L/W
PSMT402,PAMT402Unit I Fourier series
Unit II Dirichlet's theorem 5 4
Unit III Fejer's theorem and applications
Unit IV Dirichlet theorem in the unit disc
Calculus on Manifolds
Course Code Unit Topics Credits L/W
PSMT403,PAMT403Unit I Multilinear Algebra
Unit II Dierential Forms 5 4
Unit III Basics of Submanifolds of Rn
Unit IV Stokes' Theorem and applications
Optional Course
PSMT404,PAMT404 OC1 :Optional Course I 2 4
OC 2: Optional Course II 2 4
Project Course
PSMT405,PAMT405 Project Course 4 4
Note:
1. PSMT401/PAMT401, PSMT402/PAMT402, PSMT403/PAMT403 are compulsory courses
for Semester IV.
2. PSMT 404/PAMT 404 is an Optional Course for Semester IV. This is a Choice Based
Course.
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3. PSMT 405/PAMT 405 is a project based Course for Semester IV. The projects for this
course are to be guided by the Faculty members of the Department of Mathematics of
the concerned college. Each project shall have maximum of 08(eight) students. The
workload for each project is 1L/W.
Teaching Pattern for Semester IV
1. Four lectures per week for each of courses: PSMT401/PAMT401, PSMT402/PAMT402,
PSMT403/PAMT403 & PSMT404/PAMT404. Each lecture is of 60 minutes duration.
In addition, there shall be tutorials, seminars as necessary for each course.
SEMESTER III
All Results have to be done with proof unless otherwise stated.
PSMT301,PAMT301 Algebra III
Unit I. Groups (15 Lectures)
Simple groups, A5is simple.
Solvable groups, Solvability of all groups of order less than 60;Nilpotent groups, Zassenhaus
Lemma, Jordan-Holder theorem,
Direct and Semi-direct products, Examples such as
(i) The group of ane transformations x7!ax+bas semi-direct product of the group of
linear transformations acting on the group of translations.
(ii) Dihedral group D2nas semi-direct product of Z2andZn:
Classication of groups of order 12:(Ref: M. Artin ,Algebra )
Unit II. Representation of nite groups (15 Lectures)
Linear representations of a nite group on a nite dimensional vector space over C:Ifis a
representation of a nite group Gon a complex vector space V;then there exits a G-invariant
positive denite Hermitian inner product on V:Complete reducibility (Maschke's theorem).
The space of class functions, Characters and Orthogonality relations. For a nite group G;
there are nitely many isomorphism classes of irreducible representations, the same number as
the number of conjugacy classes in G:Two representations having same character are isomor-
phic. Regular representation. Schur's lemma and proof of the Orthogonality relations. Every
irreducible representation over Cof a nite Abelian group is one dimensional.
Character tables with emphasis on examples of groups of small order.
Reference for Unit II:
1.M. Artin ,Algebra , Prentice Hall of India.
2.S. Sternberg ,Group theory and Physics , Cambridge University Press,.
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Unit III. Modules (15 Lectures)
Modules over rings, Submodules. Module homomorphisms, kernels. Quotient modules. Isomor-
phism theorems. (ref: D.S. Dummit and R.M. Foote ,Abstract Algebra )
Generation of modules, nitely generated modules, (internal) direct sums and equivalent
conditions. (ref: D.S. Dummit and R.M. Foote ,Abstract Algebra )
Free modules, free module of rank n:For a commutative ring R;Rnis isomorphic to Rmif
and only ifn=m:Matrix representations of homomorphisms between free modules of nites
ranks. (Ref: N. Jacobson ,Basic Algebra , Volume 1.)
Dimension of a free module over a P.I.D. (ref: S. Lang ,Algebra ).
Unit IV. Modules over PID (15 Lectures)
Noetherian modules and equivalent conditions. Rank of an R-module. Torsion submodule
Tor(M)of a module M;torsion free modules, annihilator ideal of a submodule. (ref: D.S.
Dummit and R.M. Foote ,Abstract Algebra )
Finitely generated modules over a PID: If Nis a submodule of free module M(over a P.I.D.)
of nite rank n;thenNis free of rank mn:Any submodule of a nitely generated module
over a P.I.D. is nitely generated. (ref: S. Lang ,Algebra )
Structure theorem for nitely generated modules over a PID: Fundamental theorem, Exis-
tence (Invariant Factor Form and Elementary Divisor Form), Fundamental theorem, Uniqueness.
Applications to the Structure theorem for nitely generated Abelian groups and linear operators.
(ref:D.S. Dummit and R.M. Foote ,Abstract Algebra )
Recommended Text Books :
1.D.S. Dummit and R.M. Foote ,Abstract Algebra , John Wiley and Sons.
2.S. Lang ,Algebra , Springer Verlag, 2004
3.N. Jacobson ,Basic Algebra , Volume 1, Dover, 1985.
4.M. Artin ,Algebra , Prentice Hall of India.
PSMT302,PAMT302 Functional Analysis
Unit I Baire spaces, Hilbert spaces (15 Lectures)
Baire spaces. Open subspace of a Baire space is a Baire space. Complete metric spaces are Baire
spaces and application to a sequence of continuous real valued functions converging point-wise
to a limit function on a complete metric space. (ref: Topology byJ.R. Munkres )
Hilbert spaces, examples of Hilbert spaces such as l2;L2(;);L2(Rn)(with no proofs).
Bessel's inequality. Equivalence of complete orthonormal set and maximal orthonormal basis.
Orthogonal decomposition. Existence of a maximal orthonormal basis. Parseval's identity.
Riesz Representation theorem for Hilbert spaces. (ref: Introduction to Topology and Modern
Analysis byG. F. Simmons )
Unit II. Normed Linear Spaces (15 Lectures)
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Normed Linear spaces. Banach spaces. Quotient space of a normed linear space. lp(1p1
spaces are Banach spaces.
Lp()(1p1 ) spaces: Holder's inequality, Minkowski's inequality, Lp()(1p1 )
are Banach spaces (ref: Royden ,Real Analysis ).
Finite dimensional normed linear spaces, Equivalent norms, Riesz Lemma and application to
innite dimensional normed linear spaces (ref: E. Keryszig ,Introductory Functional Analysis
with Applications ).
Unit III. Bounded Linear Transformations (15 Lectures)
Bounded linear transformations, Equivalent characterizations. The space B(X;Y ):Complete-
ness of B(X;Y )whenYis complete. Hahn-Banach theorem, dual space of a normed linear
space, applications of Hahn-Banach theorem.
Reference for unit II: E. Keryszig ,Introductory Functional Analysis with Applications ).
Unit IV. Basic Theorems (15 Lectures)
Open mapping theorem, Closed graph theorem, Uniform boundedness Principle (ref: E. Keryszig ,
Introductory Functional Analysis with Applications ).
Separable spaces, examples of separable spaces such as lp(1p<1):If the dual space X0of
Xis separable, then Xis separable (ref: B.V. Limaye ,Functional Analysis ).
Dual spaces of lp(1p <1)(ref: E. Keryszig ,Introductory Functional Analysis with
Applications )
Dual ofLp()(1p<1)spaces: Riesz-Representation theorem for Lp()(1p<1)spaces
(ref:Royden ,Real Analysis ).
Recommended Text Books :
1.G. F. Simmons ,Introduction to Topology and Modern Analysis , Tata MacGrahill.
2.B.V. Limaye ,Functional Analysis , Wiley Eastern.
3.Royden ,Real Analysis , Macmillian.
4.E. Keryszig ,Introductory Functional Analysis with Applications , Wiely India.
5.J.R. Munkres ,Topology , Pearson.
PSMT303,PAMT303 Dierential Geometry
Unit I. Isometries of Rn(15 Lectures)
Orthogonal transformations of Rnand Orthogonal matrices. Any isometry of Rnxing the
origin is an orthogonal transformation. Any isometry of Rnis the composition of an orthogonal
transformation and a translation. Orientation preserving isometries of Rn:
Re ection map about a hyperplane WofRnthrough the origin: Let Wbe a vector subspace
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ofRnof dimension n1:Letnbe any unit vector in Rnorthogonal to W:DeneT(v) =
v2hv;nin;(v2Rn):ThenTis an orthogonal transformation of Rn;andTis independent
of the choice of n:Any isometry of Rnis the composition of at most n+ 1 many re ections.
Isometries of the plane: Rotation map of R2about any point pofR2;re ection map of R2
about any line lofR2:Glide re ection of R2( obtained by re ecting about a line land then
translating by a non-zero vector vparallel tol). Any isometry of R2is a rotation, a re ection,
a glide re ection, or the identity.
References for Unit I:
1.S. Kumaresan ,A Course in Riemannian geometry .
2.M. Artin ,Algebra , PHI.
Unit II. Curves (15 Lectures)
Regular curves in R2andR3;Arc length parametrization, Signed curvature for plane curves,
Curvature and torsion of curves in R3and their invariance under orientation preserving isometries
ofR3:Serret-Frenet equations. Fundamental theorem for space curves in R3:
Unit III. Regular Surfaces (15 Lectures)
Regular surfaces in R3;Examples. Surfaces as level sets, Surfaces as graphs, Surfaces of rev-
olution. Tangent space to a surface at a point, Equivalent denitions. Smooth functions on
a surface, Dierential of a smooth function dened on a surface. Orientable surfaces. Mobius
band is not orientable.
Unit IV. Curvature (15 Lectures)
The rst fundamental form. The Gauss map, the shape operator of a surface at a point, self-
adjointness of the shape operator, the second fundamental form, Principle curvatures and direc-
tions, Euler's formula, Meusnier's Theorem, Normal curvature. Gaussian curvature and mean
curvature, Computation of Gaussian curvature, Isometries of surfaces, Covariant dierentiation,
Gauss's Theorema Egragium (statement only), Geodesics.
Recommended Text Books :
1.M. DoCarmo ,Dierential geometry of curves and surfaces , Dover.
2.C. Bar,Elementary Dierential geometry , Cambridge University Press, 2010.
3.A. Pressley, ,Elementary Dierential Geometry , Springer UTM.
4.M. Artin ,Algebra , PHI.
5.S. Kumaresan ,A Course in Riemannian geometry .
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PSMT304,PAMT304 & PSMT305,PAMT305 ELECTIVE COURSES I & II
The Elective Courses I and II will be any TWO of the following list of ten courses:
1. Algebraic Topology
Unit I. Fundamental Group (15 Lectures)
Homotopy. Path homotopy. The fundamental group. Simply connected spaces. Covering
spaces. Path lifting and homotopy lifting lemma. Fundamental group of the circle.
Unit II. Fundamental group, Applications (15 Lectures)
Deformation retracts and homotopy types. Fundamental group of Sn:Fundamental group of
the projective space. Brower xed point theorem. Fundamental theorem of algebra. Borsuk-
Ulam theorem. Seifert-Van Kampen Theorem (without proof). Fundamental group of wedge
of circles. Fundamental group of the torus.
Unit III. Covering Spaces (15 Lectures)
Equivalence of covering spaces. The lifting lemma. Universal covering space. Covering trans-
formations and group actions. The classication of covering spaces.
Unit IV. Simplicial Homology (15 lectures)

-Complexes, Simplicial Homology, computation of simplicial Homology groups for S2;T2:
Recommended Text Books:
1.James Munkres ,Topology , Prentice Hall of India, 1992.
2.Alan Hatcher ,Algebraic Topology, Cambridge University Press , 2002.
3.John Lee ,Introduction to Topological Manifolds , Springer GTM, 2000.
4.James Munkres ,Elements of Algebraic Topology , Addison Wesley, 1984.
2. Advanced Complex Analysis
Unit I. Monodromy (15 Lectures)
Holomorphic functions of one variable. Germs of holomorphic functions. Analytic continuation
along a path. Examples including z1=nandlog(z);Homotopy between paths. The monodromy
theorem.
Unit II. Riemann Mapping Theorem (15 Lectures)
Uniform convergence, Ascoli's theorem, Riemann mapping theorem.
Unit III. Elliptic Functions (15 Lectures)
Lattices in C:Elliptic functions (doubly periodic meromorphic functions) with respect to a lattice.
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Sum of residues in a fundamental parallelogram is zero and the sum of zeros and poles (counting
multiplicities) in a fundamental parallelogram is zero, Weierstrass P-function, Relation between
PandP0;Theorem that PandP0generate the eld of elliptic functions.
Unit IV. Zeta Function (15 Lectures)
Gamma and Riemann Zeta functions, Analytic continuation, Functional equation for the Zeta
function.
Recommended Text Books :
1.S. Lang ,Complex Analysis , Springer .
2.John Conway , Functions of one complex variable, Narosa India.
3.E.M. Stein and R. Shakarchi , Complex Analysis, Princeton University Press.
3. Commutative Algebra
Unit I. Basics of rings and modules (15 Lectures)
Basic operations with commutative rings and modules, Polynomial and power series rings, Prime
and maximal ideals, Extension and contractions, Nil and Jacobson radicals, Chain conditions,
Hilbert basis theorem, Localization, Local rings, Nakayama's lemma, Tensor products.
Unit II. Primary decomposition (15 Lectures)
Associated primes, Primary decomposition.
Unit III. Integral Extensions (15 Lectures)
Integral extensions, Going up and going down theorems, The ring of integers in a quadratic
extension of rationals, Noether normalization, Hilbert's nullstellensatz.
Unit IV. Dedekind Domains (15 Lectures)
Artinian rings, Discrete valuation rings, Alternative characterizations of discrete valuation rings,
Dedekind domains, Fractional ideals, Factorization of ideals in a Dedekind domain, Examples.
Recommended Text Books :
1.S. Lang ,Complex Analysis , Springer .
2.D.S. Dummit and R.M. Foote , Abstract Algebra,John Wiley and Sons.
3.M.F. Atiyah and I.G. McDonald , Introduction to Commutative Algebra, Addison
Wesley.
4. Algebraic Number Theory
Unit I. Number Fields (15 Lectures)
Field extensions, Number elds, Algebraic numbers, Integral extensions, Ring of integers in a
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number eld, Fractional ideals, Prime factorization of ideals, Norm of an ideal, Ideal classes,
The class group, The group of units.
Unit II. Quadratic Reciprocity (15 Lectures)
The Legendre symbol, Jacobi symbols, The laws of quadratic reciprocity.
Unit III. Quadratic Fields: Factorization (15 Lectures)
Quadratic elds, Real and imaginary quadratic elds, Ring of integers in a quadratic eld, The
group of units, Ideal Factorization in a quadratic eld, Examples: The ring of Gaussian integers,
The ring Z[p
5];Factorization of rational primes in quadratic elds.
Unit IV. Imaginary Quadratic Fields: The Class Group (15 Lectures)
The Ideal class group of a quadratic eld, Class groups of imaginary quadratic elds, The
Minkowski lemma, The niteness of the class group, Computation of class groups, Application
to Diophantine equations.
Recommended Text Books :
1.P. Samuel ,Algebraic Theory of Numbers , Dover, 1977.
2.M. Artin ,Algebra , Prentice-Hall, India, 2000.
3.Marcus ,Number Fields , Springer.
4.Algebraic Number Theory , T.I.F.R. Lecture Notes, 1966.
5. Partial Dierential Equations
Unit I. Classication of second order Linear partial dierential equations (15
Lectures)
Review of the theory of rst order partial dierential equations, Method of Characteristics
for Quasilinear First Order Partial Dierential Equations, The general Cauchy problem, Cauchy-
Kowalevsky theorem, Local solvability: the Lewy example.
The classication of second order linear partial dierential equations.
Unit II. Laplace operator (15 Lectures)
Symmetry properties of the Laplacian, basic properties of the Harmonic functions, the Funda-
mental solution, the Dirichlet and Neumann boundary value problems, Green's function. Appli-
cations to the Dirichlet problem in a ball in Rnand in a half space of Rn:Maximum Principle
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for bounded domains in Rnand uniqueness theorem for the Dirichlet boundary value problem.
Unit III. Heat operator (15 Lectures)
The properties of the Gaussian kernel, solution of initial value problem ut
u= 0 forx2
Rn&t > 0andu(x;0) =f(x)(x2Rn):Maximum principle for the heat equation and
applications.
Unit IV. Wave operator (15 Lectures)
Wave operator in dimensions 1, 2 & 3; Cauchy problem for the wave equation. DAlemberts
solution, Poisson formula of spherical means, Hadamards method of descent, Inhomogeneous
Wave equation.
Recommended Text Books :
1.F. John ,Partial Dierential Equations , Narosa publications.
2.G.B. Folland ,Introduction to partial dierential equations , Prentice Hall.
6. Numerical Analysis
Unit I. Basics of Numerical Analysis (15 Lectures )
Representation of numbers: Binary system, Hexadecimal system, octal system. Ones comple-
ment, twos complement in binary application for subtraction. Russian Peasants method for
multiplication and its application in binary system for multiplication.
Errors in numerical computation of numbers: Floating point representation of numbers, rounding
o errors and Mantissa & exponent, Truncation errors, Inherent errors.
Stability, Dierence between stability and convergence in numerical methods. Numerically un-
stable methods, ill conditioned problems and illustrations of the concepts by examples. Errors
in series approximation.
Unit II. Numerical linear algebra (15 Lectures )
Gauss elimination to obtain LU factorization of matrices and partial pivoting in matrices.
Gauss-Jacobi and Gauss-Siedel methods for solving system of linear equations with derivation
of convergence.
Greschgorin theorem and Brower's theorem for bounds of eigenvalues of matrices.
Unit III. Roots of equations (15 Lectures )
Only the methods listed below are expected.
Bisection method with proof of convergence and derivation of and rate of convergence, Regula
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Falsi and secant methods, Newton-Raphson method. Rates of convergence, sucient condition
for convergence of iteration scheme and application to Newton-Raphson method.
Ramanujan's method, Muller's method for detection of complex roots, Berge-Vieta and Bairstwo
methods for roots of polynomials.
Unit IV. Numerical Integration (15 Lectures )
Lagrange's interpolation formula, uniqueness of interpolation, general error in interpolation (No
other interpolation formulae expected).
Trapezoidal and Simpson's1
3-rule in composite forms, Gauss Legendre numerical integration,
Gauss-Chebeschev numerical integration, Gauss-Hermite numerical integration, Gauss-Laguree
numerical integration with the derivation of all methods using the method of undetermined
coecients.
Estimation of error in numerical integration by using using error constant method as in [1]. Only
the following seven methods are expected using C and C++ Programs: 1) Bisection method 2)
Newton-Raphson method 3) Gauss-Jacobi method 4) Gauss-Siedel method 5) Trapezoidal rule
6) Simpson's rule 7) Muller's method.
Recommended Text books :
1.M.K Jain, S.R.K. Iyengar and R.K. Jain ,Numerical methods for Scientists and
Engineers , New Age International, Fifth Edition or next editions.
2.S.S.Sastry ,Numerical Methods , Prentice-Hall India.
3.V. Rajaram ,Computer Oriented Numerical Methods , Prentice Hall India.
4.H.M. Antia ,Numerical Analysis , Hidustan Pulications.
7. Graph Theory
Unit I. Connectivity (15 Lectures)
Overview of Graph theory-Denition of basic concepts such as Graph, Subgraphs, Adjacency and
incidence matrix, Degree, Connected graph, Components, Isomorphism, Bipartite graphs etc.,
Shortest path problem-Dijkstra's algorithm, Vertex and Edge connectivity-Result 0;
Blocks, Block-cut point theorem, Construction of reliable communication network, Menger's
theorem.
Unit II. Trees (15 Lectures)
Trees-Cut vertices, Cut edges, Bond, Characterizations of Trees, Spanning trees, Fundamental
cycles, Vector space associated with graph, Cayley's formula, Connector problem- Kruskal's
algorithm, Proof of correctness, Binary and rooted trees, Human coding, Searching algorithms-
BFS and DFS algorithms.
Unit III. Eulerian and Hamiltonian Graphs (15 Lectures)
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Eulerian Graphs- Characterization of Eulerian Graph, Randomly Eulerian graphs, Chinese post-
man problem- Fleury's algorithm with proof of correctness. Hamiltonian graphs- Necessary
condition, Dirac's theorem, Hamiltonian closure of a graph, Chvatal theorem, Degree majorisa-
tion, Maximum edges in a non-hamiltonian graph, Traveling salesman problem.
Unit IV. Matching and Ramsey Theory (15 Lectures)
Matchings-augmenting path, Berge theorem, Matching in bipartite graph, Halls theorem, Konig's
theorem, Tutte's theorem, Personal assignment problem, Independent sets and covering- +=
p;Gallai's theorem, Ramsey theorem-Existence of r(k;l);Upper bounds of r(k;l);Lower bound
formr(k;l)2m=2wherem= minfk;lg;Generalize Ramsey numbers- r(k1;k2;

;kn);Graph
Ramsey theorem, Evaluation of r(G;H )when for simple graphs G=P3;H=C4:
Recommended Text Books :
1.J.A. Bondy and U.S.R. Murty ,Graph Theory with Applications , Elsevier.
2.J. A. Bondy and U.S. R. Murty ,Graph Theory , GTM 244 Springer, 2008.
3.M. Behzad and A. Chartrand ,Introduction to the Theory of Graphs , Allyn and
Becon Inc., Boston, 1971.
4.K. Rosen ,Discrete Mathematics and its Applications , Tata-McGraw Hill, 2011.
5.D.B.West ,Introduction to Graph Theory , PHI, 2009.
8. Design Theory
Unit I. Introduction to Balanced Incomplete Block Designs (15 Lectures)
What Is Design Theory? Basic Denitions and Properties, Incidence Matrices, Isomorphisms
and Automorphisms, Constructing BIBDs with Specied Automorphisms, New BIBDs from Old,
Fishers Inequality.
Unit II. Symmetric BIBDs (15 Lectures)
An Intersection Property, Residual and Derived BIBDs, Projective Planes and Geometries, The
Bruck-Ryser-Chowla Theorem. Finite ane and and projective planes.
Unit III. Dierence Sets and Automorphisms (15 Lectures)
Dierence Sets and Automorphisms, Quadratic Residue Dierence Sets, Singer Dierence Sets,
The Multiplier Theorem, Multipliers of Dierence Sets, The Group Ring, Proof of the Multiplier
Theorem, Dierence Families, A Construction for Dierence Families.
Unit IV. Hadamard Matrices and Designs (15 Lectures)
Hadamard Matrices, An Equivalence Between Hadamard Matrices and BIBDs, Conference Ma-
trices and Hadamard Matrices, A Product Construction, Williamson's Method, Existence Results
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for Hadamard Matrices of Small Orders, Regular Hadamard Matrices, Excess of Hadamard Ma-
trices, Bent Functions.
Recommended Text Books :
1.D. R. Stinson ,Combinatorial Designs: Constructions and Analysis , Springer,2004.
2.W.D. Wallis ,Introduction to Combinatorial Designs , (2nd Ed), Chapman & Hall.
3.D. R. Hughes and F. C. Piper ,Design Theory , Cambridge University Press, Cam-
bridge, 1985.
4.T. Beth, D. Jungnickel and H. Lenz ,Design Theory , Volume 1 (Second Edition),
Cambridge University Press, Cambridge, 1999.
9. Coding Theory
Unit I. Error detection, Correction and Decoding (15 Lectures)
Communication channels, Maximum likelihood decoding, Hamming distance, Nearest neigh-
bor/minimum distance decoding, Distance of a code.
Unit II. Linear codes (15 Lectures)
Linear codes: Vector spaces over nite elds, Linear codes, Hamming weight, Bases of linear
codes, Generator matrix and parity check matrix, Equivalence of linear codes, Encoding with
a linear code, Decoding of linear codes, Cossets, Nearest neighbour decoding for linear codes,
Syndrome decoding.
Unit III. Cyclic codes (15 Lectures)
Denitions, Generator polynomials, Generator and parity check matrices, Decoding of cyclic
codes, Burst-error-correcting codes.
Unit IV. Some special cyclic codes (15 Lectures)
Some special cyclic codes: BCH codes, Denitions, Parameters of BCH codes, Decoding of
BCH codes.
Recommended Text Books :
1.San Ling and Chaoing xing ,Coding Theory- A First Course .
2.Lid and Pilz ,Applied Abstract Algebra , 2nd Edition.
10. Integral Transforms
Unit I. Laplace Transform (15 Lectures)
Denition of Laplace Transform, Laplace transforms of some elementary functions, Properties of
Laplace transform, Laplace transform of the derivative of a func- tion, Inverse Laplace Transform,
Properties of Inverse Laplace Transform, Inverse Laplace Transform of derivatives, Convolution
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Theorem, Heaviside's expansion theorem, Application of Laplace transform to solutions of ODEs
and PDEs.
Unit II. Fourier Transform (15 Lectures)
Fourier Integral theorem, Properties of Fourier Transform, Inverse Fourier Transform, Convolu-
tion Theorem, Fourier Transform of the derivatives of functions, Parseval's Identity, Relationship
of Fourier and Laplace Transform, Application of Fourier transforms to the solution of initial
and boundary value problems.
Unit III. Mellin Transform (15 Lectures)
Properties and evaluation of Mellin transforms, Convolution theorem for Mellin transform, Com-
plex variable method and applications.
Unit IV. Z-Transform (15 Lectures)
Denition of Z-transform, Inversion of the Z-transform, Solutions of dierence equations using
Z-transform.
Recommended Text Books :
1.Brian Davies ,Integral transforms and their Applications , Springer.
2.L. Andrews and B. Shivamogg ,Integral Transforms for Engineers ,Prentice Hall of
India.
3.I.N.Sneddon ,Use of Integral Transforms , Tata-McGraw Hill.
4.R. Bracemell ,Fourier Transform and its Applications , MacDraw hill.
Skill Course
The Skill Course is any one of the following three courses:
Skill Course I: Business Statistics
Unit I. Data Classication, Tabulation and Presentation (15 Lectures)
Classication of Data: Requisites of Ideal Classication, Basis of Classication.
Organizing Data Using Data Array: Frequency Distribution, Methods of Data Classication,
Bivariate Frequency Distribution, Types of Frequency Distributions.
Tabulation of Data: Objectives of Tabulation, Parts of a Table, Types of Tables, General and
Summary Tables, Original and Derived Tables.
Graphical Presentation of Data: Functions of a Graph, Advantages and Limitations of Diagrams
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(Graph), General Rules for Drawing Diagrams.
Types of Diagrams: One-Dimensional Diagrams, Two-Dimensional Diagrams, Three-Dimensional
Diagrams, Pictograms or Ideographs, Cartograms or Statistical Maps.
Exploratory Data Analysis: Stem-and-Leaf Displays.
Unit II. Measures of Central Tendency (15 Lectures)
Objectives of Averaging, Requisites of a Measure of Central Tendency, Measures of Central
Tendency,
Mathematical Averages: Arithmetic Mean of Ungrouped Data, Arithmetic Mean of Grouped
(Or Classied) Data, Some Special Types of Problems and Their Solutions, Advantages and
Disadvantages of Arithmetic Mean, Weighted Arithmetic Mean.
Geometric Mean: Combined Geometric Mean, Weighted Geometric Mean, Advantages, Disad-
vantages and Applications of G.m.
Harmonic Mean: Advantages, Disadvantages and Applications of H.M. Relationship Between
A.M., G.M. and H.M.
Averages of Position: Median, Advantages, Disadvantages and Applications of Median.
Partition Valuesquartiles, Deciles and Percentiles: Graphical Method for Calculating Partition
Values.
Mode: Graphical Method for Calculating Mode Value. Advantages and Disadvantages of Mode
Value.
Relationship Between Mean, Median and Mode, Comparison Between Measures of Central Ten-
dency.
Unit III. Measures of Dispersion (15 Lectures)
Signicance of Measuring Dispersion (Variation):Essential Requisites for a Measure of Variation.
Classication of Measures of Dispersion.
Distance Measures: Range, Interquartile Range or Deviation.
Average Deviation Measures: Mean Absolute Deviation, Variance and Standard Deviation,
Mathematical Properties of Standard Deviation, Chebyshev's Theorem, Coecient of Varia-
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tion.
Unit IV. Skewness, Moments and Kurtosis (15 Lectures)
Measures of Skewness: Relative Measures of Skewness.
Moments: Moments About Mean, Moments About Arbitrary Point, Moments About Zero
or Origin, Relationship Between Central Moments and Moments About Any Arbitrary Point,
Moments in Standard Units, Sheppard's Corrections for Moments.
Kurtosis: Measures of Kurtosis.
Reference Book :
J.K. Sharma ,Business Statistics , Pearson Education India, 2012.
Skill Course II: Statistical Methods
Unit I. Basic notions of Statistics (15 Lectures)
Measures of central tendencies: Mean, Median, Mode.
Measures of Dispersion: Range, Mean deviation, Standard deviation. Measures of skewness.
Measures of relationship: Covariance, Karl Pearson's coecient of Correlation, Rank Correlation.
Basics of Probability.
Reference: Chapter 8 of the book: C. R. Kothari and G. Garg ,Research Methodology
Methods and Techniques , New Age International.
Unit II. Sampling and Testing of Hypothesis (15 Lectures)
Sampling Distribution, Student's t-Distribution, Chi-square ( 2) Distribution, Snedecor's F-
Distribution. Standard Error. Central Limit theorem. Type I and Type II Errors, Critical
Regions. F-test, t-test, 2test, goodness of Fit test.
Reference for Unit II: Chapter 9,10 and 11 of the book: C. R. Kothari and G. Garg ,
Research Methodology Methods and Techniques , New Age International.
Unit III. Analysis of Variance (15 Lectures)
The Anova Technique. The basic Principle of Anova. One Way ANOVA, Two Way ANOVA.
Latin square design. Analysis of Co-variance.
Reference for Unit III: Chapter 12 of the book: C. R. Kothari and G. Garg ,Research
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Methodology Methods and Techniques , New Age International.
Unit IV. Use of package R (15 Lectures)
Ras Statistical software and language, methods of Data input, Data accessing, usefull built-in
functions, Graphics with R, Saving, storing and retrieving work.
Reference for Unit IV: Chapter 1 of the book: S.G. Purohit, S.D. Gore and S.R. Desh-
mukh ,Statistics using R , Narosa.
Skill Course III: Computer Science
Aim: Mathematics students are well versed in logic.This Skill course aims at giving input of
necessary skills of alogirtms and data structures and relational database background so that the
studetnts are found suitable to be absorbed as trainee software professional in industry.
Prerequisite for this course: Good knowledge of C , C++ or java or python.
Unit I. OOPS Concepts (15 lectures)
Basics of object oriented programming principles, templates, reference operators NEW and delete
in C++, the java innovation which avoids use of delete, classes polymorphism friend functions,
inheritrance, multiple inheritance operator overloading basiscs only references for UNit I: [3], [4]
Unit II. Basic Algorithms (15 lectures)
Basic algorithms, selection sort, quick sort, heap sort, priory queses, radix sort, merge sort,
dynamic programming, app pairs, shortest paths, image compression, topological sorting, single
source shortest paths reference, hashing intuitive evaluation of running time.
references:[1], [2]
Unit III. Data Structures (15 lectures)
Stacks queues, linked lists implementation and simple applications, trees implementation and
tree traversal ( stress on binary trees),
Unit IV. Relational Databases (15 lectures)
concept of relational databases , normal forms BCNF and third normal forms. Armstrongs ax-
ioms. Relatiional algebra and operations in it.
references: [5]
recommended Text Books :
1.T. Aron and others ,Dtat structure using C .
2.S. Sahani ,Data tructures and applications , TMH.
3.Balaguruswamy ,Programming in C++ .
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4.Programming in Java , Schaum Series.
5.J.D. Ullam ,Principles of Database systems .
SEMESTER IV
PSMT401,PAMT401 Field Theory
Unit I. Algebraic Extensions (15 lectures)
Revision: Prime subeld of a eld, denition of eld extension K=F; algebraic elements, minimal
polynomial of an algebraic element, extension of a eld obtained by adjoining one algebraic
element.
Algebraic extensions, Finite extensions, degree of an algebraic element, degree of a eld exten-
sion. Ifis algebraic over the led Fandm(x)is the minimum polynomial of overF;then
F()is isomorphic to F[X]=(m(x)):IfFKLare elds, then [L:F] = [L:K][K:F]:
IfK=F is a eld extension, then the collection of all elements of Kwhich are algebraic over F
is a subeld of K:IfL=K;K=F are algebraic extensions, then so is L=F: Composite led K1K2
of two subelds of a eld and examples. (Ref: D.S. Dummit and R.M. Foote ,Abstract
Algebra )
Classical Straight-edge and Compass constructions: denition of Constructible points, lines,
circles by Straight-edge and Compass starting with (0;0)and(1;0);denition of constructible
real numbers. If a2Ris constructible, then ais an algebraic number and its degree over Q
is a power of 2:cos 20is not a constructible number. The regular 7-gon is not constructible.
The regular 17-gon is constructible. The Constructible numbers form a subeld of R:Ifa>0
is constructible, then so ispa:(Ref: M. Artin ,Algebra , Prentice Hall of India)
Impossibility of the classical Greek problems: 1) Doubling a Cube, 2) Trisecting an Angle, 3)
Squaring the Circle is possible. (Ref: D.S. Dummit and R.M. Foote ,Abstract Algebra )
Unit II. Normal and Separable Extensions (15 lectures)
Splitting eld for a set of polynomials, normal extension, examples such of splitting elds of
xp2(pprime), uniqueness of splitting elds, existence and uniqueness of nite elds. Algebraic
closure of a eld, existence of algebraic closure.
Separable elements, Separable extensions. In characteristic 0;all extensions are separable. Frobe-
nius automorphism of a nite eld. Every irreducible polynomial over a nite eld is separable.
Primitive element theorem.
Reference for Unit II: D.S. Dummit and R.M. Foote ,Abstract Algebra )
Unit III. Galois Theory (15 Lectures)
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Galois group G(K=F )of a eld extension K=F; Galois extensions, Subgroups, Fixed elds,
Galois correspondence, Fundamental theorem of Galois theory.
Unit IV. Applications (15 Lectures)
Cyclotomic eld Q(n)(splitting eld of xn1overQ), cyclotomic polynomial, degree of
Cyclotomic eld Q(n):D.S. Dummit and R.M. Foote ,Abstract Algebra )
Galois group for an irreducible cubic polynomial, Galois group for an irreducible quartic polyno-
mial. (Ref: M. Artin ,Algebra , Prentice Hall of India)
Solvability by radicals in terms of Galois group and Abel'stheorem on the insolvability of a
general quintic. (Ref: D.S. Dummit and R.M. Foote ,Abstract Algebra )
Rcommended Text Books :
1.D.S. Dummit and R.M. Foote ,Abstract Algebra , John Wiley and Sons.
2.M. Artin ,Algebra , Prentice Hall of India, 2011.
Additional Reference Books:
1.S. Lang ,Algebra , Springer Verlag, 2004
2.N. Jacobson ,Basic Algebra , Dover, 1985.
PSMT402,PAMT402 Fourier Analysis
Unit I. Fourier series (15 Lectures)
The Fourier series of a periodic function, Dirichlet kernel, Bessel's inequality for a 2-periodic
Riemann integrable function, convergence theorem for the Fourier series of a 2-periodic and
piecewise C1- function, uniqueness theorem (If f;g are2-periodic and piecewise smooth func-
tion having same Fourier coecients, then f=g).
Relating Fourier coecients of fandf0wherefis continuous 2-periodic and piecewise C1-
function and a convergence theorem: If fis continuous 2-periodic and piecewise C1-function,
then the Fourier series of fconverges to fabsolutely and uniformly on R:
Reference for Unit I : Sections 2.1, 2.2, 2.3 of the G.B. Folland ,Fourier Analysis and
its Applications , American Mathematical Society, Indian Edition 2010.
Unit II. Dirichlet's theorem (15 Lectures)
Review: Lebesgue measure of R;Lebesgue integrable functions, Dominated Convergence theo-
rem, Bounded linear maps (no questions be asked).
Denition of Lebesgue integrable periodic functions (i.e. L1-periodic), Fourier Coecients of
L1-periodic functions, L2-periodic functions. Any L2-periodic function is L1-periodic. Riemann-
Lebesgue Lemma (if fis Lebesgue integrable periodic function, then lim
jnj!1^f(n) = 0):The
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Converse of Riemann-Lebesgue lemma does not hold (ref: W. Rudin ,Real and Complex
Analysis , Tata McGraw Hill). Bessel's inequality for a L2-periodic function.
Dirichlet's Theorem on point-wise convergence of Fourier series (If fis Lebesgue integrable
periodic function that is dierentiable at a point x0;then the Fourier series of fatx0converges
tof(x0)) and convergence of the Fourier series of functions such as f(x) =jxjon[;]:
Reference for Unit II : Sections 13A, 13B, 13C of R. Beals ,Analysis An Introduction ,
Cambridge University Press, 2004.
Unit III. Fejer's Theorem and applications (15 Lectures)
Fejer's Kernel, Fejer's Theorem for a continuous 2-periodic function, density of trigonometric
polynomials in L2(;);Parseval's identity.
Convergence of Fourier series of an L2-periodic function w.r.t the L2-norm, Riesz-Fischer the-
orem on Unitary isomorphism from L2(;)onto the sequence space l2of square summable
complex sequences.
Reference for Unit III : Sections 13D, 13E, 13F of R. Beals ,Analysis An Introduction ,
Cambridge University Press, 2004.
Unit IV. Dirichlet Problem in the unit disc (15 Lectures)
Laplacian, Harmonic functions, Dirichlet Problem for the unit disc, The Poisson kernel, Abel
summability, Abel summability of periodic continuous functions, Weierstrass Approximation
Theorem as application, Solution of Dirichlet problem for the disc.
Applications of Fourier series to Isoperimetric inequality in the plane and Heat equation on the
circle.
Reference for Unit IV :E. M. Stein and R. Shakarchi ,Fourier Analysis an Intro-
duction , Princeton University Press, 2003.
PSMT403,PAMT403 Calculus on Manifolds
Unit I. Multilinear Algebra (15 Lectures)
Multilinear map on a nite dimensional vector space VoverR;andk-tensors onV;the collection
Tk(V)(or
k(V)) of allk-tensors on V;tensor product S
TofS2Tk(V) &T2Tk(V);
Alternating tensors and the collection ^kVof of allk-tensors on V;The exterior product (or
wedge product, basis of ^kV;orientations of a nite dimensional vector space VoverR:
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Reference : pp 75-84 in Chapter 4 of M. Spivak ,Calculus on Manifolds , W.A. Benjamin.
Unit II. Dierential Forms (15 Lectures)
Dierential forms: k-forms on Rn;wedge product !^of ak-form!andl-form;the exterior
derivative and properties, Pull back of forms and properties, Closed and exact forms, Poincare's
lemma.
Reference : pp 86-97 in Chapter 4 of M. Spivak ,Calculus on Manifolds , W.A. Benjamin.
Unit III. Basics of submanifolds of Rn(15 Lectures)
Submanifolds of Rn;submanifolds of Rnwith boundary, Smooth functions dened on Subman-
ifolds of Rn;Tangent vectors and Tangent spaces of Submanifolds of Rn:
p-forms and dierentiable p-forms on a submanifold of Rn;exterior derivative d!of any dieren-
tiablep-form on a submanifolds of Rn;Orientable submanifolds of Rnand Oriented submanifolds
ofRn;Orientation preserving maps, Vector elds on submanifolds of Rn;outward unit normal
on the boundary of a submanifold of Rnwith non-empty boundary, induced orientation of the
boundary of an oriented submanifold of Rnwith non-empty boundary.
Reference : pp 109-122 in Chapter 4 of M. Spivak ,Calculus on Manifolds , W.A. Benjamin.
Unit IV. Stokes's Theorem (15 Lectures)
IntegralR
[0;1]kwof ak-form on the cube [0;1]k;IntegralR
cwof ak-form on an open subset
AofRkwherecis a singular k-cube inA:Theorem (Stokes' Theorem for k-cubes): If!is a
(k1)-form on an open subset AofRkandcis a singular k-cube inA;thenR
cd!=R
@cw:
(ref: pp 100-108 in Chapter 4 of M. Spivak ,Calculus on Manifolds , W.A. Benzamin.)
Integration of a dierentiable k-form on an oriented k-dimensional submanifold MofRn:
Change of variables theorem: If c1;c2: [0;1]k!Mare two Orientation preserving maps in M
and!is anyk-form onMsuch that!= 0 outside ofc1([0;1]k)\c2([0;1]k);thenR
c1!=R
c2!;Stokes' theorem for submanifolds of Rk;Volume element, Integration of functions on a
submanifold of Rk;Classical theorems: Green's theorem, Divergence theorem of Gauss, Green's
identities. (ref: pp 122-137 in Chapter 4 of M. Spivak ,Calculus on Manifolds , W.A. Benjamin
Inc.)
Additional Reference Books :
1.V. Guillemin and A. Pollack ,Dierential Topology , AMS Chelsea Publishing,
2010.
2.J. Munkres ,Analysis on Manifolds , Addision Wesley.
3.A. Browder ,Mathematical Analysis , Springer International edition.
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PSMT404,PAMT404 Optional Course
This course has two parts: First part is the Optional Course I & the second part is Optional
Course II.
Optional Course I: Linear Programming
Unit I. Linear Programming (15 Lectures)
Operations research and its scope, Necessity of operations research in industry, Linear program-
ming problems, Convex sets, Simplex method, Theory of simplex method, Duality theory and
sensitivity analysis, Dual simplex method.
Unit II.Transportation Problems (15 Lectures)
Transportation and Assignment problems of linear programming, Sequencing theory and Trav-
elling salesperson problem.
Optional Course II: Optimisation
Unit III. Unconstrained Optimization (15 Lectures)
First and second order conditions for local optima, One-Dimensional Search Methods: Golden
Section Search, Fibonacci Search, Newtons Method, Secant Method, Gradient Methods, Steep-
est Descent Methods.
Unit IV. Constrained Optimization Problems (15 Lectures)
Problems with equality constraints, Tangent and normal spaces, Lagrange Multiplier Theorem,
Second order conditions for equality constraints problems, Problems with inequality constraints,
Karush-Kuhn-Tucker Theorem, Second order necessary conditions for inequality constraint prob-
lems.
Recommended Text Books :
1.H.A. Taha ,Operations Research-An introduction , Macmillan Publishing Co. Inc., NY.
2.K. Swarup, P. K. Gupta and Man Mohan ,Operations Research , S. Chand and
sons, New Delhi.
3.S.S. Rao ,Optimization Theory and Applications , Wiley Eastern Ltd, NewDelhi.
4.G. Hadley ,Linear Programming , Narosa Publishing House, 1995.
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5.F.S. Hillier and G.J. Lieberman ,Introduction to Operations Research (Sixth Edi-
tion), McGraw Hill
6.Chong and Zak ,Introduction to Optimization , Wiley-Interscience, 1996.
7.Rangarajan and K. Sundaram ,A First Course in Optimization Theory , Cam-
bridge University Press.
Scheme of Examination
I. Semester End Theory Examinations :
The scheme of examination for the revised course in the subject of Mathematics for Semesters
III & IV at the M.A./M.Sc. Programme (CBCS) will be as follows.
There shall be a Semester-end external Theory examination of 100 marks for all the courses of
Semester III and IV- except for the project courses USMT405/UAMT405 -to be conducted by the
University.
Theory Question paper pattern:
a) There shall be ve questions each of 20 marks.
b) On each unit there will be one question and the fth one will be based on entire syllabus.
c) All questions shall be compulsory with internal choice within each question.
d) Each question may be subdivided into sub-questions a, b, c, and the allocations of marks
depend on the weightage of the topic.
e) Each question will be of 30 marks when marks of all the subquestions are added (including
the options) in that question.
II. Evaluation of Project work
( courses: USMT405/UAMT405):
The evaluation of the Project submitted by a student shall be made by a Committee appointed
by the Head of the Department of Mathematics of the respective college.
The presentation of the project is to be made by the student in front of the committee appointed
by the Head of the Department of Mathematics of the respective college. This committee shall
have two members, possibly with one external referee. Each project output shall be displayed
on the website of the University.
The Marks for the project are detailed below:
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Contents of the project : 40 marks
Presentation of the project : 30 marks
Viva of the project : 30 marks.
Total Marks= 100 per project per student.
III. Evaluation of Skill Course
At the end of Semester III, there shall be a Semester end Internal Examination of 100 marks for
the evaluation of the Skill Course and students shall be given grades A;B;C;D (A being the
highest grade and D being the lowest grade). A student shall be required to get minimum of C
grade to qualify for the M.A./M.Sc. degree CBCS Programme in the subject of Mathematics.
The marks of the Skill Course shall not be considered for the calculation of CGPA score of the
M.A./M.Sc. degree.
??????
26