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3rd August 2015, 09:42 AM
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Join Date: Apr 2013
Re: Phd Syllabus of Kumaun University Nainital

Kumaun University was founded in 1973, it is a state university and It has two campuses, in Nainital and Almora,……

Applicants must have a Master’s degree of the Kumaun University (hereafter referred to as University) or of any other University

Ph.D mathematics syllabus:
1. Real Analysis: Riemann integrate functions; improper integrate, their convergence and
uniform convergence. Eulidean space R¯ , Boizano – Weleratrass theorem, compact.
Subsets of R•, Heine – Borel theorem, Fourier series.Continuity of functions on R”,
Differentiability of F: R• > Rm, Properties of differential, partial and directional
derivatives, continuously differentiable functions. Taylor’s series. Inverse function
theorem, implicit function theorem. Integral functions, line and surface integrals, Green’s
theorem. Stoke’s theorem.

2. Complex Analysis: Cauchy’s theorem for convex regions, Power series representation of
Analytic functions. Liouville’s theorem, Fundamental theorem of algebra, Riemann’s
theorem on removable singularities, maximum modulus principle. Schwarz lemma, Open
Mapping theorem, Casoratti–Weierstrass–theorem, Weierstrass’s theorem on uniform
convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions,
Riemann Surfaces.

3. Advanced Analysis: Elements of Metric Spaces, Convergence, continuity, compactness,
Connectedness, Weierstrass’s approximation Theorem. Completeness, Bare category
theorem, Labesgue measure, Labesgue integral, Differentiation and integration

4. Algebra: Symmetric groups, alternating groups, Simple groups, Rings, Maximal ideals,
Prime ideals, integral domains, Euclidean domains, principal ideal domains, Unique
Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations,
Reduction of matrices to Canonical Forms, Inner Product Spaces, Orthogonality, quadratic
Forms, Reduction of quadratic forms.

5. Numerical analysis: Finite differences, interpolation ; Numerical solution of algebric
equation; Iteration; Newton–Rephson method; Solution on linear system; Direct method;
Gauss elimination method; Matrix–Inversion, elgenvalue problems; Numerical
differentiation and integration.Numerical solution of ordinary differential equation;
iteration method, Picard’s method, Euler’s method and improved Euler’s method.

6. Linear Programming Basic Concepts: Convex sets. Linear Programming Problem
(LPP). Examples of LPP, Hyperplane, open and closed half – spaces. Feasible, basic
feasible and optimal solutions. Extreme point and graphical method. Simplex method,
Duality in linear programming. Transformation and assignment problems.

7. Measure Theory: Measurable and measure spaces; Extension of measures, signed
measures, Jordan – Hahn decomposition theorems. Integration, monotone convergence
theorem, Fatou’s lemma, dominated convergence theorem. Absolute continuity. Radon
Nikodym theorem, Product measures, Fubini’s theorem.

8. Functional Analysis: Banach Spaces, Hahn – Banach Theorem, Open mapping and
closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity
of Linear Transformations. Dual Space, Embedding in the second dual, Hilbert Spaces,
Projections. Orthonormal Basis, Riesz – representation theorem. Bessel’s inequality,
parsaval’s identity, self adjoined operators, Normal Operators.

9. Ordinary and partial differential equations: Introduction to differential equations,
Linear Equations with Variable Coefficients, Euler’s Method, The Existence and
Uniqueness Theorem, Homogeneous equations with constant coefficients, Fundamental
solutions, linear independence, Wronskian, Non-homogeneous equations: method of
undetermined coefficients, Non-homogeneous equations: method of variation of
parameters, Partial differential equations, Monge’s method, Canonical forms,
Characterization of a partial differential equation, Heat equation, Wave equation,
Laplace’s equation

10. Topology: Topological spaces, open sets, closed sets, neighbourhoods, Inetrior, exterior
and boundary of sets. Metric spaces, Induced Topology, Complete metric spaces,
compactness in metric spaces, Continuity and homeomorphism, Connectedness,
Compactness, Countability and separability.

For more detailed syllabus here I am attaching pdf file which is free for download……

Kumaun University
Sleepy Hallow
Nainital, Uttarakhand 263001

Attached Files
File Type: doc Phd Syllabus of Kumaun University Nainital.doc (84.5 KB, 221 views)

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