# 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

**Syllabus Semester -I **

**M. Sc. Mathematics**

**Paper – I **

**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

**Syllabus Semester -I **

**M. Sc. Mathematics**

**Paper – II **

**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

**Syllabus Semester -I **

**M. Sc. Mathematics**

**Paper – III**

### 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

**Syllabus Semester -I **

**M. Sc. Mathematics**

**Paper – IV**

### 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

**Syllabus Semester -I **

**M. Sc. Mathematics**

**Paper – V**

**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.