Syllabus covered in Book
Homomorphism theorems on groups, conjugate elements, classes and class equation of a finite group, sylows Theorem, Cauchy’s theorem for finite Abelian group. Normal and subnormal series, composition series, Jordan-Holder Theorem, Solvable groups. Ideal, Principal Ideal rings, Division and Euclidean algorithms for polynomials over a field, Euclidean rings and domains. Unique factorization theorems, unique factorization domains, Finite Field extension. Algebraic and Transcendental extensions, seperable and Inseparable extensions, Normal Extensions, Perfect field.
Riemann-Stieltjes integral, Properties of Integral and Differentiation, Point wise and uniform convergence of sequence & series of functions, Cauchy critrerion, Weirstass M-test, Abel and Dirchlet test for Uniform Convergence, Uniform Convergence and continuity. Measurable sets, Lebesgue outer measure and measurability. Measurable functions, Borel and Lebesgue measurability, Non measurable sets. Convergence of sequence of measurable functions, Lebesgue integral of a bounded function.
Partial Differential Equations
Existence and uniqueness of solution of (dy/dx)=f(X,Y). Example of PDE Classifcation. Cannonical forms, Nonlinear Forms, Nonliner First order PDE-Complete integrals, Envelopes. Method of solving Second order PDE-separation of variable and Cauchy’s problem.
Laplace’s Equation, Heat Equation and Wave Equation upto three dimension in polar coordinates, their fundamental solutions by variable separation.
Calculus of Varriations, Shortest distance, Minimum surface of revolution, Brachistochrone problem, Isoperimetric problem, Geodesic.
Syllabus:- The hypergeometric functions: The Gauss’ hypergeometric function F (a,c,z) Its integral form, continuous function relations, the hypergeometric differential equation, elementary properties, simple and quardratic transformations, Gauss and Kummar’s theorems.
The Generlised by pergeometric function (a1,……..ap,b1……….by;z) its diffential equations, continuous function relations, integral forms, saalschut’s, Whipple’s, Dixon’s theorems, contour Integral representations.
Bessel function, Its differential equation, pure and differential recurrence relations, generating function, modified Bessel function and its properties.
Confluent hypergeometric function (a.b;z) definitions, properties, recurrence relations, kummar’s formulas.
Generating Functions: Generating Functions of the form G (2xt-t2) sets generated by etA (t) exp(-xt/(1-t) and the related theorems.
Sets and Proposition: Cardinality, Mathematical Induction, Principle of inclusion and exclusion pigeon hole principle.
Logic, Predicate, Validity of Statements, Quantification, Proof of Implications/Identities, Mehtod of proofs.
Boolean Algebra: Boolean function and expression, Propositional calculus.
Design and implementation of Digital networks, Application to switching and Logic circuits
Graph Theory, Lattces.