## M.Phil. Mathematics: Syllabus

M.Phil. Mathematics is one to two years course and has 2 to 4 semesters depending on the colleges. While famous colleges like Delhi University, JNU, Jamia Islamia University have a two years degree with a 4-semester program, while Amity University of Applied Sciences, Kurukshetra Univeroffersoffer students with one-year degree program with two semesters.

The first part requires students to choose 4 coursework out of the offered courses and they are evaluated at the end of the First semester.

The Second year or second semesters for a one year program comprises purely of a dissertation prepared by the student through research. Thus, a topic of research is chosen as per the student’s interest.

The course delivery methods in the program include discussions, presentations, classroom performance, assignments, attendance and seminars.

#### Scheme of Assessment

In M.Phil. Mathematics, the scheme of assessment is based on the written test, the internal assessment, viva-voce as well as dissertation which comprises more than half of the marks. The students are examined out of 300 marks for the entire coursework.

The final weight age for each of these parameters is given below:

#### Part I

Examination | Marks Allotted |
---|---|

Theory | 75 |

Internal Assessment | 25 |

#### Part II

Examination | Marks Allotted |
---|---|

Dissertation | 150 |

Viva-Voce | 50 |

### Syllabus

The syllabus for M.Phil Mathematics in India follows a pattern set by UGC ascribed to all the universities. An overview of the syllabus is given below:

Name of Paper | Topics | Description |
---|---|---|

Hydrodynamic and Hydromagnetic Stability-I | Analysis in terms of normal modes, Non-dimensional number, Benard Problem, Basic hydrodynamic equations, Boussinesq approximation, Perturbation equations, Principle of exchange of stabilities, Equations governing the marginal state, Exact solution when instability sets in as stationary convection for two free boundaries. , Thermal instability in rotating fluid. , Perturbation equations. Variational Principle for stationary convection, On the onset of convection as overstability; the solution for the case of two free boundaries. Thermal instability in the presence of a magnetic field. , Perturbation equations. Rayleigh-Taylor instability. Perturbation equations. Inviscid case. , | This course offers an analysis of different modes of Hydrodynamicic and hydromagnatic stability. Also, it offers a study in the effect of rotation and effect of the vertical magnetic field through variable illustrative examples. |

Hydrodynamic and Hydromagnetic Stability-II | Initiation of Magnetoconvection , Magnetohydrodynamic simple Bénard instability problem , Reformulation of the Simple Bénard and Thermohaline Instability Problem , Modified Analysis of Simple Bénard instability problem and thermohaline instability problem , The eigenvalue problem , Limitations of the Complex Wave Velocity in the Instability Problem of Heterogeneous Shear , The perturbation equations , | It is the thorough evaluation of the characteristics of the marginal state and the marginal state solution through practical works. |

Fluid Flow Instability, MHD, Plasmas and Geophysical Fluid Dynamics | Fluid Flow Instability , Kelvin-helmohlotz instability , Perturbation equations and boundary conditions , MHD and Plasmas , Introduction of Magnetohydrodynamics (MHD) , Geophysical Fluid Dynamics , Porosity , Methods for measurement of porosity , Darcy’s law , Darcy’s Oberbeck-Boussinesq (DOB) equations for material. , | This course allows a study of different equations pertaining to the flow of stagnant,magnatic and porous media. |

Groups, Rings and Modules | GROUPS & Ideals , Characters of finite abelian groups , The Character group, the orthogonality relations for characters , Maximal Ideal , Generators , Basic Properties of Ideals , Algebra of Ideals , Quotient Rings , Ideal in Quotient Rings, Local Rings , The Jacobson Radical Moludes; The radical of a ring , Artinian rings , Semisimple Artinian rings , The density theorem , Semisimple rings , Applications of Wedderburn’s theorem. , | It introduces to the Analytic Number Theory and study of theorems and its applications. |

Matrix Analysis | Matrices , Partioned Matrices , Special types of matrices, Change of basis , Unitary Matrices , Unitary equivalence, Schur’s unitary triangularization theorem, Normal Matrices , Definitions, Properties and characterizations of Hermitian Matrices , Canonical forms , The Jordan Canonical form: proof observations and applications. , Variational characterization of eigenvalues of Hermitian Matrices , Positive definite matrices, Definitions and properties, characterization The polar form and the singular value decomposition , Genaralised inverse , Inequalities for the positive definite matrices , | It offers the in-depth knowledhe regarding the variable relations between Normal, Unitary and Hermitian Matarices along with implications of Schur’s theorem. |

Boundary Layer Theory | Methods of boundary-Layer Control , Boundary-Layer suction , Injection of a different gas (Binary boundary Layers) , Boundary –Layer equations. , The Method of successive approximations. , C.C.Lin’s Method for periodic external flows. , Boundary-Layer formation after the impulsive start of motion , Boundary-Layer formation in accelerated motion , Periodic boundary-layer flows , Parameter perturbation , | Study of the boundary layer flow and its equations with different gases. |