Special functions – Cylindrical and spherical functions in mathematical physics – Geometric functions – Transform methods – Complex Fourier transform – Wiener Hopf techniques with applications on wave and radiation problems – Integral formulas – Dirichlet Problem – Green's function – Potential problems. .
Solution methods for a large system of equations – Iterative techniques – Conjugate gradient method – Matrix eigenvalue problem – Jacobi's method – Householder's method – Given's method – Minimization theory – Numerical solution of ordinary differential equations – Stability study - Numerical solution of partial differential equations – Finite differences – Finite elements methods .
Local and global maxima and minima – Concave and convex functions – One dimensional examples – Fibonacci search – Golden section search – Descent methods – Conjugate direction methods – Quasi Newton methods – Kuhn Tucker optimality conditions – Lagrange multipliers – Quadratic programming – Penalty and barrier methods – Geometric programming – Separable problems
State space concepts – State equation – Modeling of physical systems – Linearity and time invariance – Discrete time systems – Stability – controllability – Observability – Liapunov stability analysis and optimal solution techniques – Approximate and optimal solution in the time and frequency domains – Transfer function – Sampling and reconstruction of signals – Transformations – Stability of discrete time systems .
Normal forms – Precedence rules for arithmetic expressions – Context free grammar – Error detection and correction codes – Code optimization.
Basic concepts of data – Linear lists – Strings , arrays and orthogonal lists – Representation of trees and graphs – Storage structures – Linked and multilinked storage structures – Data storage organization technique – Internal and external organization – Sorting and comparison methods – Search in scattered tables .
Advanced topics in system simulation and optimal solutions .
Advanced topics in computer systems and system programming .
Differential equations – Integral equations – Linear algebra – Numerical analysis – Probability and statistics – Partial differential equations – Graph theory – Discrete mathematics.
Random variables and probability distributions – Statistical independence – Moments and generating functions – Conditional probability – Maximum likelihood estimation – Hypothesis tests – Regression and correlation.
Systems of linear differential equations – Vector analysis – Tensor analysis – Fourier Integral and its application for solving ordinary and partial differential equations – Calculus of variations – Boundary value problems and Green's function.
Laplace transform – Series solution of differential equations – Special functions including Bessel and Legendre functions – Functions of complex variable – Evaluation of real integrals by the method of residues.
Definitions and axioms – Comparison between the Euclidean geometry and the parabolic , elliptic and hyperbolic geometries.
Representation of the geometrical elements – Position and
metric problems – Central affinity – Solids – Polyhedron –
Circle – Sphere – Cone – Cylinder – Drawing the perception
on a vertical picture plane by the metric points method and
by the vanishing point method
.
Indexed projection: Representation of the geometrical elements – Problems of position – Metric problems – Solids – Topographic surfaces – Slopes of excavation and filling. Axonometric projection: System of reference - Representation of the elements – Straight line – Plane – Polyhedrons – Circle - Sphere – Cone – Cylinder – Surfaces of revolution. Inclined axonometric projection.
Helix: Helical curves - Helical surfaces
.
Introduction to high dimensional geometry – Analytical discussion of the elements of the high dimensional spaces – System of reference – Representation of the geometrical elements - Problems of position - Metric problems – Representation of polytopes – Hypersphere – Hypercone – Hypercylinder .
Vectors – Multiplication and products – Differentiation – Dependence – Plane and space curves – Tangents – Arc length – Osculating plane – Moving trihedron – Torsion – Surfaces – Tangent plane – Normal line – Area – First, second and third fundamental forms – Angles – Geodesics – Principal and Gaussian curvature – Mapping and transformation of surfaces – Conformal mapping – Mercator – Stereographic maps – Isometric mapping – Developable surfaces – Equal maps – Lambert's maps.
Axioms and incidence – Homogeneous coordinates – Some fundamental projective theorems – Pappus theorem – Disargue's theorem – Principle of duality – Cross ratio – One to one correspondence – Ranges and pencils – Harmonic pencils – Involution – Conics analytically treated – Degenerate conic – Pole and polar line – Parametric form of conic -
Isotropic lines
.
Laplace's equation – Green's functions – Complex variable methods – Sturm Liouville problem and eigenfunction expansion – Hilbert space methods for elliptic equations – Existence – Uniqueness – Regularity.
Normed linear space – Banach space – Hilbert space – Distribution theory .
Euclidean and metric spaces – Series – Differentiability – Riemann Stieltjes integral – Sequences and series of functions – Measure and integration – Lebesgue integral – Fubini's theorem – Lp spaces.
Markov chains – Queuing theory – Reliability theory – Information theory and coding.
Volterra integral equations – Resolvent kernel – Euler integrals – Fredholm equations of the second kind – Iterated kernels – Degenerate kernel – Approximate methods of solving integral equations.
Detailed study of different kinds of integral transforms with their properties and applications .
Existence and uniqueness of solutions – Linear systems with constant , periodic and analytic coefficients – Singularities of autonomous systems – Self adjoint eigenvalue problem – Expansion in terms of eigenfunctions – Stability theory and Liapunov functions.
Special functions in the real and complex domains – Bessel functions – Legendre polynomials – Hermite polynomials, etc. – Applications.
Infinite products – Entire functions – Analytic continuation – Riemann surfaces.
Selected topics on the application of the fuzzy theory in mathematics – Boolean algebra and Li algebra.
Topics selected by the advisors.
Topics selected by the advisors.
Introduction (Scaling and wavelet equations – Bases and frames – Time , frequency and scale ) – Filter banks ( Ideal reconstruction – Polyphase matrix ) – Orthogonal filter banks (Paraunitary matrices – Orthogonal filters – Half band filters – Spectral analysis – Daubechies filters ) – Multiresolution analysis (Wavelets from filter banks – Infinite product formula ) – Wavelet theory (Successive method for the dilation equation – Relationship between the smoothness of the scaling function and wavelets – some applications on wavelets).
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