Set theory. Real numbers. Inequalities. Straight lines and circles. Trigonometry. Complex Numbers.
Set theory. Real numbers. Inequalities. Straight lines and circles. Complex Numbers. Systems of linear equation, Elementary matrices and their applications.
Limits, Continuous functions. The derivative, formulas of differentiation, differentials. Related rates, Extrema. Rolle’s and mean value theorems. Graph sketching Optimization. Indefinite integrals, definite integrals, fundamental theorem of calculus. Applications of definite integrals to find area and volume.
Logarithmic and exponential functions. Inverse trigonometric and hyperbolic functions. Techniques of integration. Indeterminate forms and improper integrals. Conic sections. Plane curves and polar coordinates.
Functions and their graphs, the derivative, applications of the derivative, simple integrations, law of growth and decay.
Trigonometry, analytical geometry in two dimensions, linear equations, functions and their graphs, derivatives and its geometric interpretation, simple integration and its applications.
Maximal ideals. Principal ideal domains, definition of unique factorization domains. Principal ideal domains are unique factorization domains. Euclidean domains. Introduction to extension fields.
Electrostatics. Dielectrics. Magnetostatics. Induced magnetism. Steady currents. Maxwell's equations. Quasi-stationary Fields. Electromagnetic waves. Notion of electric charges.
The basic idea of mathematical modeling of real world problems. Modeling using graphs; proportionality; fitting, and optimization, experimental modeling. Dimensional analysis and similitude, simulation modeling; modeling using the derivative, interactive dynamic systems.
Pairing. Stirling's formula. Recurrence relations and generating functions. The inclusion exclusion principle, Boolean functions. Switching circuits. Applications to digital computers.
Graphs and digraphs. Connectivity. Eulerian and Hamiltonian problems. Trees. Planarity. Matrix representation. Shortest path problem. Colouring. Applications.
The fundamental theorem of arithmetic. Congruences. The theorems of Fermat and Wilson. Quadratic residues. The quadratic reciprocity law. Diophantine equations.
Topology of the real number system. Functions. Limits of functions. Limit theorems for functions. Continuity. Types of discontinuity. Continuity theorems. The derivative. Roll's and the mean value theorems. Riemann integral. Properties of the definite integral. Sequences of real numbers. Theorems on sequences. Infinite series. Continuity of functions of several variables.
The algebra of complex numbers. Analytic functions, integration of complex functions. Cauchy's theorem and the Cauchy integral formula. Power series, Laurent series, Calculus of residues.
Rectilinear motion of a particle with variable acceleration. Simple harmonic motion. Damped and forced oscillations. Motion of a particle in a plane using Cartesian, polar and intrinsic coordinates. Central orbits. Kepler's laws. Moments and products of intertia. Kinetic energy and angular momentum of rigid body. The general equations of motion. Two-dimensional problems. Generalized coordinates and Lagrange equations of motion. Small oscillations.
Homomorphism. Isomorphism theorems. Class equation. Cauchy theorem for finite groups. Direct sums and Jordan Holder theorem. Structure of groups of orders up to 8. Integral domains. Division rings. Subrings. Ideals. Fields of quotients. Polynomial rings. Eienstein criteria.
Permutation groups, conjugate classes in Sn, the simplicity of An, Direct products, the fundamental theorem of finite abelian groups. The Sylow theorems, groups of order 2p, pq and orders up to 10. Jordan-Holder theorem, solvable groups.
Numerical solution of nonlinear equations. Numerical solution of linear algebraic systems: direct and iterative methods. Polynomial interpolation. Least-squares approximation. Numerical integration.
Vector calculus, Asymptotic expansions, Calculus of variations, Applications of ODEs to sciences.
Equation of Laplace, Fourier and Hankel transforms, Boundary value problems, applications.
Equations of motion. Two-dimensional motion. Streaming motions. Sources and sinks. Theorem of Schwarz and Christoffel. Stoke’s stream function and three-dimensional motion.
Systems of linears equations: direct methods: elimination and its matrix analysis, special structured systems, Thomas and Choleski algorithms, iterative methods: Jacobi and Gauss-Seidel, SOR, optimal SOR. The linear least-squares problem: normal equations, QR-and SVD-decompositions, error analysis. Matrix eigenvalue problem: power method, deflation, Given and Householder orthogonal transformations, the QR-algorithm, reduction to Hessenberg form; eigenvalue problem for symmetric matrices.
Axioms for the Euclidean plane, transformations in the plane, distance, isometries, dilatations, similarities, inversions. Projective plane. Desargues’ theorem. Harmonic sequences. Pappus theorem. Homogeneous coordinates.
Metric spaces: open and closed sets, continuous functions, product of metric spaces, complete metric spaces. Topological spaces: open and closed sets, continuous functions, product of topological spaces, connectedness, compactness, separation and countability axioms.
Sequences and series of functions, the Weierstrass approximation theorem, Picard's existence theorem and equicontinuous families of functions. Lebesgue measure and the Lebesgue integral of bounded and unbounded functions. Convergence theorems. Fourier series.
MATH 403 - Advanced Linesr Algebra: Linear transformations, change of bases matrices, dual spaces, irreducible polynomials, vector spaces as F[x]
modules, characteristic polynomial decomposition companion matrices, rational and Jordan canonical forms
Sets and functions in R^n, continuity and compactness, differentiability, Jacobian matrix and gradient, inverse and implicit function theorems, extrema, Hessian matrix, Taylor’s formula, integration in R^n, vector fields, integral theorems.
Polynomial rings, field extensions, finite fields, computations in finite fields, irreducible polynomials. Linear machines, shift registers, codes over GF(g), Hamming codes, cyclic codes, BC codes, Reed-Solomon codes, encoding cyclic codes, BCH decoding as an FSR problem, algorithmic BCH decoding (only illustrations).
Topological spaces, product spaces, quotient spaces, convergence, separation axioms, first and second countable spaces, compactness, connectedness, fundamental groups.
Analytic projective geometry. Affine, hyperbolic and elliptic geometries.
Review of tensor analysis. The principle of special relativity. Lorentz transformations. minkowski geometry. Relativistic mechanics and electrodynamics. Introduction to the general theory of relativity. The principle of equivalence, metric in a gravitational field, Einstein's law of gravitation. Shwarzchild solution.
Analysis of stress and strain. Principal axes. Compatibility equations. Generalised Hooke's law.Beltrami-Michell equations. The fundamental boundary value problems. Applications.
First and second order linear partial differential equations. The method of characteristics. Separation of variables and its application to linear partial differential equations. Fourier series. Sturm-Liouville theory, Green’s functions.
Selection of mathematical models based on differential equations and linear systems. Analysis of models, development of solution algorithms and sensitivity of models to perturbations in data. Computer simulation for visualization and interpretation of physical phenomenon governed by these models.
Fourier transform and inversion techniques, applications in signal processing. Laplace and Mellin transforms. Discrete linear systems and filters, discrete Fourier transforms. Fast Fourier transform. Perturbation methods.
Numerical integration.Some basic rules. Gaussian and Romberg integration. Numerical solution of ordinary differential equations. Initial value problems. One step and multistep methods (convergence, absolute stability). Boundary value problems.
Numerical solution of partial equations selected topics to be chosen from: Theory of approximation, variational methods, Eigenvalue problems, further linear algebra, integral equations.
Curves in space. Frenet-Serrt formulas. Curvature and torsion. Surfaces. First and second fundamental forms. Intrinsic geometry.
Metric spaces, Properties of normed spaces, Inner product spaces. Bounded and continuous linear operators, Linear functionals, Hilbert adjoint operators, Fundamental theorems for normed and Banach spaces.
Finite incidence structures. Incidence matrices. Block design. Projective and affine planes. Coolineations. Coordination and ternary rings. The Bruck-Ryser theorem.
Propositional calculus. First order logic. First order recursive arithmetic. Arithmatization of syntax. The incompleteness theorem and other applications. |

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