Difference equations. Numerical solution of ordinary differential equations: Initial value problems: Numerical stability, Taylor series method, Euler and modified Euler methods and stability analysis, Runge-Kutta methods, Multistep methods, Predictor-Corrector method, convergence and stability. System of ordinary differential equations. Boundary Value Problems: Shooting and direct methods.
Viscous and inviscid fluid, the Navier-Stokes equation of motion, rotational and irrational flows. Theory of surface wave: Equation of Motion, Wave Terminology, Analytical solution of the wave problem, Dispersion relation of the wave motion, Classification of water waves, Particle motion and Pressure, Superposition of waves, Wave reflection and standing wave, Wave energy and group velocity, Wave Refraction, Wave Diffraction. Finite amplitude waves: Mathematical formulation, Perturbation method of solution.
Linear and Nonlinear diffraction theory. Nonlinear equations: autonomous and non-autonomous systems, phase portrait, stability of equilibrium points, Lyapunov exponents, periodic solutions, local and global bifurcations, Poincare-Bendixon theorem, Hartmann-G robmann theorem, Center Manifold theorem.
Nonlinear oscillations: perturbations and the Kolmogorov-Arnold-Moser theorem, limit cycles. Chaos: one-dimensional and two-dimensional Poincare maps, attractors, routes to chaos, intermittency, crisis and quasi periodicity. Synchronization in coupled chaotic oscillators.
Rings and Ideals ;Rings and ring homomorphism , Ideals, quotient rings, Zero divisors, Nil potent Element ,Units, Prime Ideals , and maximal ideals, Nilradical and Jacobson radical, Operation on ideals, Extension and contraction ; Modules ; modules and modules homomorphism ,sub module ,Direct sum product of modules, restriction and extension of scalars , Exactness properties of the tensor product ,algebras ,tensor product of algebra ;Rings and modules of fraction,L ocal Properties Extended and contracted ideals in the rings of fractions, Primary decomposition, Integral dependence, The going up theorem Integrally closed integral domains, Valuation Topologies and completion, Filtrations, Graded Rings and modules, The associated graded ring , Artin Ress Theorem , Dimension Theory, Hilbert function , Dimensions theory of Noetherian local rings, Regular local rings, Transcendental dimension, Depth :M-Regular Sequences, Cohen Macaulay Rings.
Operators on Hilbert spaces: Bounded linear operator on Hilbert spaces, spectrum of an operator, weak , norm and strong operator topologies, normal, self adjoint, unitary and compact operator and their spectra. Diagonalization, spectral theorem and applications: diagonalization for a compact self adjoint operator, spectral theorem for compact norm operator, spectral calculus, application to strum-Liouville problem.
Positive operators : positive linear maps of finite dimensional space and their norms, Schur products, completely positive maps.
Sigma- rings, sigma algebra, measurable space, countability and sub — addivity of a measure, Borel measure, Lebesgue outer measure, measurable sets, construction of non-measureable set, the contor set. Measurable functions and their properties, almost everywhere property, approximation of measurable functions with the simple measurable functions and step functions.
Singed measures, absolute continues functions and their properties, singular measures , Radon — Nikodym theorem with applications. We prove that sharply dominating Archimedean atomic lattice effect algebras can be characterized by the property called basic decomposition of elements. As an application we prove the state smearing theorem for these effect algebras.
Compact orthoalgebras. We initiate a study of topological orthoalgebras TOAs , concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice.
Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated is of finite height.
We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated is determined by that of its space of atoms. Orthocomplete effect algebras. Invariant subspaces for a linear transformation, simultaneous triangulation and diagonalization, Jordan decomposition of a linear transformation.
Rational and Jordan canonical forms of matrices. Inner product spaces, The Gram-Schmidt orthogonalization. The adjoint of a linear operator, normal and self-adjoint operators, Unitary and orthogonal operators, orthogonal projections and spectral Theorem. Recommended Reading M.
Hoffman and R. Kunze, Linear Algebra , Pearson Education Nathan Jacobson, Basic Algebra Vol. I, Dover Publications Luthar and I. Passi, Algebra Vol.
System of first order linear equations, general solution of homogeneous linear systems, fundamental matrix, non-homogeneous linear systems, linear systems with constant coefficients. First order partial differential equations PDEs , linear and quasi-linear equations, general first order PDE for a function of two variables. Second order PDEs, characteristic for linear and quasi-linear second order equations. The Cauchy problem, Cauchy-Koualevski theorem. The method of separation of variables, the Laplace equation, the heat equation, the wave equation.
Recommended Reading Earl A. Shepley L. Ross, Differential Equations , Wiley Ravi P. Metric spaces, neighbourhoods and continuity. Topological spaces, open sets, closed sets, examples, order topology, subspace topology, product topology, quotient topology. Limit points, convergence of nets in topological spaces. Tychonoff theorem, one point compactification, Stone-Cech compactification.
Recommended Reading Paul R. Armstrong, Basic Topology , Springer Complex differentiation, analytic functions, polynomials, power series, exponential and trigonometric functions. Cauchy- Riemann equations, analytic functions as mappings, exponential function, logarithm, harmonic functions. Zeros of analytic functions, index of a closed curve.
Poles and essential singularities, Casorati-Weierstrass theorem. Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof theorems. Additional Topics Gamma function, Riemann zeta function, prime number theorem. Analytic continuation, spaces of analytic functions and of meromorphic functions. Riemann mapping theorem, infinite products, Weierstrass factorization Theorem.
Recommended Reading Lars V.