सबसे पहले तो आप सब का धन्यवाद जो आप मेरे ब्लॉग के साथ जुड़े हुए हो।सबसे पहले में आपको HP UNIV में MATH के पेपर कुछ बात बता चाहता हूँ आशा है यह बातें आप सब को बहुत मदत करेंगी। M.A/M.SC. MATH को HP UNIV ने दो साल की डिग्री कोर्स को चार भागों में बांटा गया है। प्रत्येक पेपर 60 अंक की होगी ओर हर पेपर को तीन खंडों में विभाजित किया जाएगा। प्रत्येक खंड में तीन प्रश्न शामिल होंगे।
HP UNIV SYLLABUS OF MSC./M.A MATH SEM 1.
Subject-M101 Real Analysis-I
Section –I
The Riemann-Stieltjes Integral
Section –I
Definition and existence of Riemann-Stieltjes integral, Properties of the Integral,
Integration and differentiation. The Fundamental theorem of calculus. Integration of vector –
valued functions. Rectifiable curves.
Section –II
Sequences and Series of Functions
Section –II
Pointwise and uniform convergence, Cauchy Criterion for uniform convergence.
Weierstrass M-Test. Abel’s and Dirichlet’s tests for uniform convergence. Uniform convergence
and continuity. Uniform convergence and Riemann – Stieltjes integration. Uniform convergence
and differentiation. Weierstrass approximation Theorem. Power series, Uniqueness theorem for
power series. Abel’s and Taylor’s Theorems.
Section –III
Section –III
Functions of Several Variables
Linear Transformations. Differentiation. Partial derivatives. Continuity of partial
derivatives. The contraction Principle. The Inverse Function Theorem. The Implicit Function
Theorem, Derivatives in an open subset of Rn
, Chain rule, Derivatives of higher orders, The
Rank Theorem. Determinants, Jacobians.
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Subject-M102 Advanced Algebra-I
Section –I
Conjugacy and G-Sets. Normal Series, Solvable Groups, Nilpotent Groups, Direct
Products, Finitely Generated Abilian Groups, Invariants of a Finite abelian Groups, Syllow
Theorems, Groups of Orders p2
, pq.
Section – II
Definition and Examples of Rings, Some Special Classes of Rings, Homomorphisms,
Ideals and Quotient Rings, More Ideals and Quotient Rings and The Field of Quotients of an
Integral Domain.
Euclidean Rings, a Particular Eudclidean Ring, Polynomial Rings, Polynomials over the
Rational Field, Polynomial Rings over Commutative Rings.
Section – III
Unitary Operators, Normal Operators, Forms on Inner Product Spaces, Positive Forms,
More on Forms, Spectral Theory
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Subject-M103 Ordinary Differential Equations
Section –I
Existence and Uniqueness Theory
Some Concepts from Real Function Theory. The Fundamental Existence and Uniqueness
Theorem. Dependence of Solutions on Initial Conditions and on the Funcition f. Existence and
Uniqueness Theorems for Systems and Higher-Order equations.
The Theory of Linear Differential Equations
Introduction. Basic Theory of the Homogeneous Linear System. Further Theory of the
Homogeneous Linear System. The Nonhomogeneous Linear System. Basic Theory of the nthOrder
Homogeneous Linear Differential Equation. The nth-Order Nonhomogeneous Linear
equation.
Section – II
Sturm-Liouville Boundary-Value Problems
Sturm-Liouville Problems. Orthogonality of Characteristic Functions. The Expansion of a
Function in a Series of Orthonormal Functions.
Strumian Theory
The separation theorem, Sturm’s fundamental theorem Modification due to Picone,
Conditions for Oscillatory or non-oscillatory solution, First and Second comparison theorems.
Sturm’s Oscillation theorems. Application to Sturm Liouville System.
Section – III
Nonlinear Differential Equations
Phase Plane, Paths, and Critical Points. Critical Points and paths of Linear Systems.
Critical Points and Paths of Nonlinear Systems. Limit Cycles and Periodic Solutions. The
Method of Kryloff and Bogoliuboff.
M104 Operations Research-I
Section –I
Hyperplane and hyperspheres, Convex sets and their properties, convex functions. Linear
Programming Problem (LPP): Formulation and examples, Feasible, Basic feasible and optimal
solutions, Extreme points. Graphical Methods to solve L.P.P., Simplex Method, Charnes Big M
Method, Two phase Method, Degeneracy, Unrestricted variables, unbounded solutions, Duality
theory, Dual LPP, fundamental properties of Dual problems, Complementary slackness, Dual
simplex algorithin, Sensivity analysis.
Section – II
Integer programming:
Gomory’s Method, Branch and Bound Method.
Transportation Problem (TP):
Mathematical formulation, Basic feasible solutions of T.Ps by
North – West corner method, Least cost-Method, Vogel’s approximation method. Unbalanced
TP, optimality test of Basic Feasible Solution (BFS) by U-V method, Stepping Stone method,
degeneracy in TP.
Assignment Problem (AP):
Mathematical formulation, assignment methods, Hungarian
method, Unbalanced AP.
Section – III
Goal programming Problem (GPP)
formulation of G.P. Graphical Goal attainment method,
simplex method for GPP.
Game theory:
Two-person, zero-sum games, The maximin – minimax principle, pure strategies,
mixed strategies, Graphical solution of 2xn and mx2 games, Dominance property, General
solution of m x n rectangular games, Linear programming problem of GP.
Network Techniques
Shortest path model, Dijkastra algorithm, Floyd’s algorithm, Minimal Spanning tree,
Maximal flow problem.
M.A./M.Sc. (Mathematics) First Semester Course
Section –I
Continuum hypothesis, Newton’s Law of Viscosity, Some Cartesian Tensor Notations,
General Analysis of Fluid Motion, Thermal Conductivity, Generalised Heat conduction.
Fundamental Equations of Motion of Viscous Fluid
Equation of State, Equation of Continuity, Navier – Stokes (NS) Equations (equation of
Motion, Equation of Energy, Streamlines & Pathlines, Vorticity and Circulation (Kelvin’s
Circulation Theorem).
Section – II
Dynamical Similarity (Reynold’s Law), Inspection Analysis- Dimensional Analysis,
Buckingham – π - Theorem, and its Applications π –products and coefficients, Non-dimensional
parameters and their physical importance.
Exact Solutions of the N S Equations
Steady Motion between parallel plates (a) Velocity distribution, (b) Temperature
Distribution, Plane Couette flow, plane Poiseuille flow, generalized plane Couette flow.
Flow in a circular pipe (Hagen-Poiseuille flow (a) velocity distribution (b) Temperature
distribution.
Section – III
Flow between two concentric Rotating Cylinders (Couette flow): (a) Velocity distribution
(b) Temperature distribution.
Flow due to a plane wall suddenly set in motion, flow due to an oscillating plane wall.
Plane Couette flow with transpiration cooling.
Steady Flow past a fixed sphere: Stokes equation and Oseen’s equation of flow.
Theory of Lubrication. Prandtl’s boundary layer equations, the boundary layer on a flat plate
(Blassius equation), Characteristic boundary layer parameters.
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