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Magadh University PhD Entrance Paper-I Mathematics Syllabus

Time: 2 Hours

Full Marks: 100 Marks

General Instructions

Ten objective questions to be set from each the five topics. Each questions carries 2 marks. T number of questions to be attempted is 50.

Basic Concepts of Real Analysis: Fin countable and uncountable sets, Bounded unbounded sets, Archimedean property, orde field, completeness of R, Extended real num system, sequences and series, lim sup and lim a sequence, continuity, uniform continuity, type discontinuities, Differentiablity, Mean value theor sequence and series of functions, unifo convergence, Riemann integral.

Complex analysis: Algebra of complex numb analytic functions, Cauchy Riemann equations, integrals, Cauchy’s theorem, Integral form power series, Morera’s theorem, Taylor’s Laurent’s series, classification of singulariti Residues, Contour integration.

Algebra: Group, subgroups, normal subgrou Quotient groups, Homomorphisms, cyclic grou Permutation groups, Cayley’s theorem, Ri ideals, Integral domains, fields, Polynomial ring.

Linear Algebra: Vector spaces, subspaces, quotient spaces, linear independence, Bases, art and dimension, The algebra of linear transformations, kernel, range, isomorphism, Matrix representation of a linear transformation, change of bases, linear functionals.

Differential equations: First order ODE, on xesingular solutions, Initial value problems of first order psin. ODE, Homogeneous and non homogeneous linear inetu ODE, variation of parameters, Lagrange’s and Charpit’s methods of solving first order PDE. PDE’s of higher orders with constant coefficients.

Magadh University PhD Entrance Paper-II Mathematics Syllabus

Time Duration: 3 Hours

Full Marks- 100

Group: A

This paper will be divided into two groups. Group nopio soso A shall contain questions from General papers of M.A./ M.Sc. carrying 50 marks and group B shall contain questions from special papers carrying 50 marks. The questions shall be short answer type, each carrying 5 marks. Group A shall comprise 14 topics and one questions shall be set from each topic. The candidate shall be required to answer 10 questions from Group A. Group B shall comprise 5 topics. Five questions shall be set from each of these five topics. The candidate shall be required to choose any two topics and answer 10 question. The questions are to be so selected that they can be easily answered within the allotted time.

1. Real Analysis: Differentiability of functions from Rn to Rm, partial derivatives, directional derivative continuously differentiable functions, Mean value theorem for functions of several variables, partial derivatives and differentiability of higher, Implicit function theorem.
2. Complex Analysis: Algebra of complex nos Analytic functions, Cauchy’s theorem and Integral formula, Power series, Taylor’s and Laurent’s series, Residues, Contour Integration.
3. Linear Algebra: Vector spaces, subspaces, quotient spaces, Linear dependence and independence, bases, dimension. The algebra of linear transformations, kernel, range, isomorphism, Matrix representation of a linear transformation, change of bases, linear functionals, dual space, eigen values and eigen vectors, Cayley – Hamilton theorem.
4. Abstract Algebra: Groups, subgroups, Normal subgroups, Quotient groups, Homomorphisms, Cyclic groups, Permutation groups, Cayley’s theorem, Rings, Ideals, Integral Domains, Fields, Polynomial Rings.
5. Differential Equations: First order ODE, singular debit solutions, Initial value problems of first order ODE, Homogeneous and non-homogeneous linear ODE, Variation of parameters, Lagrange’s and Charpit’s methods of solving first order PDE . PDE’s of higher order with constant co-efficients.
6. Differential Geometry : Space curves, their curvature and torsion, Serret – Frenet formula, mano Fundamental theorem of space curves, curves of ero surfaces, First and second Fundamental form, Gaussian curvature, Principal directions and principal curvatures.
7. Topology: continuity, Elements of topological space, convergence, Homeomorphism, compactness, connectedness, separation axioms, Fist and second countability, product spaces, Quotient sub spaces.
8. Functional Analysis: Banach space, Hahn- Banach snl to theorem, open mapping and closed graph theorems, Istren Principle of uniform boundedness, Boundedness and to foulcontinuity of linear transformations, Dual space, 210ar Hilbert space, Projection, Orthonormal bases, self- adjoint and normal operators.
9. Mechanics: Generalized co-ordinates, Lagrange’s moits equation, Hamilton’s canonical equation, variational principles, Hamilton’s principle and principle of least action, Euler’s dynamical equation of motion of rigid body.
10. Fluid Dynamics: Equation of continuity in fluid motion, Euler’s equations of motion for perfect fluids. Two dimensional motion, complex potential, Motion of sphere in perfect liquid, vorticity.

11. Integral Transforms: Laplace Transform Transform of elementary functions, Transform dervatives, Inverse Transform, Convolution theorem Fourier transform, Sine and cosine transforms
    one Inverse Fourier transform.
12. Discrete Mathematics: Partially ordered sets, Lattices, Complete Lattices, Distributive lattices, Complements, Boolean Algebra, Boolean traic expressions, Application to switching circuits.
13. Tensor Calculus: Vactor spaces and their dual, Bases and dual bases, Transformation of bases, rucovariant and contravariant vectors, Tensor of the Ales type (r,s), Tensor product of vectors, Fundamental algebraic operations of tensors, Inner product of vectors, contraction, Fundamental tensors, Christoffel symbols and their elementary properties, spn covariant differentiation of covariant vector, Brocontravariant vector and tensor, Transformation formula of Christoffel symbols.
14. Operational Research : Definition and scope of operational research, Different types of models, beh Replacement models and sequencing theory. Inventory problems and their analytical structure, simple deterministic stochastic models of Inventory.

GROUP – B

1. General Topology: Nets and filters, T2, T, and le completely regular space, Urysohn’s lemma,
   Metrization theorem, Extension theorem, Stone cech compactification, Local finiteness, Nagata – why Smirnov metrization theorem, Paracompact spaces, Weierstrass approximation theorem, The Stone Weierstrass theorem, Extended Stone – Weierstrass theorem, Uniform space.
   2.51 Functional Analysis: Construction of a topology to for a linear space, convex sets, Minkowski ble functionals, seminorms and topology defined by a oits family of semi norms, The conjugate space of a Elle Hilbert space, Adjoint and self-adjoint operators, a’ni normal operators, completely continuous operators, eto Projection operators, Banach algebra.
2. Differential Geometry: Gauss’s formulae for r,12 г12, 22, Gauss characteristic equation, Mainardi – Codazzi relation, spherical representation of a surface, Minimal surface and its properties, Nature of asymptotic lines, null lines and lines of curvature on a minimal surface, Ruled surface and its 001 fundamental magnitudes, Line of striction, Nature of tangent plane at a point, Bonnet’s theorem, eqy asymptotic lines on a ruled surface. due
3. Modern Algebra: Group complexes and subgroups, eat Isomorphism theorems, Product theorems. decomposition of a group relative to two such.

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