# RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020 | RTMNU M.Sc. Revised Sem 1 Syllabus 2020 | Nagpur University M.Sc. Syllabus Part 1 | RTMNU First Year M.Sc. Syllabus

RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020 RTM Nagpur University M.Sc Semester New Revised Syllabus is given below for Downloading. The students can Download the respective Syllabus from following given details. Just go through the given links & read the given syllabus carefully. Nagpur University First Year New Semester Online Detail syllabus given below.

## RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### Algebra -I

#### Unit 1:

Permutation Group. Group of Symmetry. Dihedral group. Commutator group. Isomorphism,Theorems. Automorphisms. Characteristic subgroup. Conjugacy and G-Sets.

#### Unit 2:

Normal Series. Solvable groups. Nilpotent groups. Cyclic decomposition of permutation group. Alternating groups. Simplicity of An.

#### Unit 3:

Direct product, semi-direct product of groups. Sylows theorems. Groups of order 2 p and pq.

#### Unit 4:

Ideals and Homomorphisms. Sum and direct sum of ideals. Maximal and prime ideals. Nilpotent and Nil ideals. Modules. Submodules. Direct sums. R-homomorphisms and quotient modules. Completely reducible modules. Free modules.

#### Text Book:

Basic Abstract Algebra :Bhattacharya, Jain, and Nagpal ,Second Edition, Cambridge University Press.

#### Reference Books:

1. Topics in Algebra, I. N. Herstein, Second Edition, John Wiley.
2. Abstract Algebra: David S.Dummit and Richard M. Foote, John Wiley.

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## RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### Real Analysis-I

#### Unit 1:

Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and differentiation. Equicontinuous families of functions. The Stone-Weierstrass theorem.

#### Unit 2:

Differentiation. The Contraction Principle. The Inverse Function Theorem. The Implicit Function Theorem. The Rank Theorem. Partitions of unity.

#### Unit 3:

The space of tangent vectors at a point of Rn. Another definition of Ta (Rn). Vector fields on open subsets of Rn. Topological manifolds. Differentiable manifolds. Real Projective space. Grassman manifolds. Differentiable functions and mappings.

#### Unit 4:

Rank of a mapping. Immersion. Sub manifolds. Lie groups. Examples of Lie groups.

#### Text Books:

1. Principles of Mathematical Analysis (Third Edition): Walter Rudin Mc GRAW – HILL Book Company.
2. An Introduction to Differentiable Manifolds and Riemannian Geometry: W. Boothby, Academic Press, 1975.

#### Reference Books:

1. Methods of Real Analysis: R.R. Goldberg, John Wiley.
2. Calculus of Several Variables: C Goffman, Harper and Row.

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## RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### Topology-I

#### Unit 1:

Countable and Uncountable sets. Examples and related Theorems. Cardinal Numbers and related Theorems. Topological Spaces and Examples.

#### Unit 2:

Open sets and limit points. Derived Sets. Closed sets and closure operators. Interior, Exterior and boundary operators. Neighbourhoods, bases and relative topologies.

#### Unit 3:

Connected sets and components. Compact and countably compact spaces. Continuous functions and

homeomorphisms.

#### Unit 4:

To and T1-spaces, T2-spaces and sequences. Axioms of countability. Separability. Regular and normal spaces.

#### Text Book:

Foundations of General Topology: W.J. Pervin, Academic press, 1964.

#### Reference Books:

1. Topology: J.R. Munkres, (second edition), Prentice Hall of India, 2002.
2. Introduction to Topology and Modern Analysis: G.F. Simmons, Mc Graw Hill 1963.
3. General Topology: J.L. Kelley, Van Nostrand, 1995.
4. Introduction to general Topology: K.D. Joshi, Wiley Eastern Ltd. 1983

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## RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### Linear Algebra and Differential Equations

#### Unit 1:

Matrices and operators, Subspaces, Bases and Dimension. Determinants, trace, and Rank. Direct sum decomposition. Real Eigen Values. Differential equations with Real Distinct Eigen values. Complex Eigen values.

#### Unit 2:

Complex vector spaces. Real operators with Complex Eigen values. Application of complexlinear algebra to differential equations. Review of topoogy in Rn. New norms for old.Exponential of operators.

#### Unit 3:.

Homogeneous linear systems. A non-homogeneous equation. Higher order systems. The primary decomposition. The S+N decomposition. Nilpotent canonical orms.

#### Unit 4:

Jordan and real canonical forms. Canonical forms and differential equations. Higher order linear equations on function spaces. Sinks and sources. Hyperbolic flows. Generic properties of operators. Significance of genericity.

#### Text Book :

Differential equations, dynamical systems and linear algebra: M.W. Hirsch and S. Smale, Academic Press, 1975.

#### Reference Book :

Dynamical systems: V.I. Arnold, Springer Verlag, 1992.

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## RTMNU M.Sc Mathematics Sem 1 New Syllabus 2020

### Integral Equations

#### Unit 1:

Preliminary concepts of integral equations. Some problems which give rise to integral equations. Conversion of ordinary differential equations into integral equations. Classification of linear integral equations. Integro-differential equations.

#### Unit 2:

Fredholm equations. Degenerate kernels. Hermitian and symmetric kernels. The Hilbert Schmidt theorem. Hermitization and symmetrization of kernels. Solutions of integral equations with Green’s function type kernels.

#### Unit 3:

Types of Voltera equations. Resolvent kernel of Voltera equations, Convolution type kernels. Some miscellaneous types of Voltera equations. Non-linear Voltera equations. Fourier integral equations. Laplace integral equations.

#### Unit 4:

Hilbert transform. Finite Hilbert transforms. Miscellaneous integral transforms. Approximate methods of solutions for linear integral equations. Approximate evaluation of Eigen values and Eigen functions.

#### Text Book:

Integral Equations: A short course: LI. G Chambers: International text book company Ltd, 1976.