Semester Offering: InterSem
 

To provide detailed mathematical background for students to understand the basic principles of Telecommunications, to obtain research insight presented in scientific papers in Telecommunications.

 

Vector Spaces, Matrices, Generalized Inverses, Linear Transformations, the Eigenvalue Problem, Functions of a Matrix, Irreducible and Monotone Matrices, Introduction of Probability Theory, Random Variables, Conditional Probability and Conditional Expectation, The Exponential Distribution and Poisson Process, Continuous-Time Markov Chains, Point-set topology, Numerical sequences and series, Real-valued functions, Complex-valued functions

 

None.

 

I. Vector Spaces

  1. Fields
  2. Vector Spaces
  3. Matrices
  4. Other Examples of Vector Spaces
  5. Subspaces
  6. Linear Dependence, Basis, and Dimension
  7. Linear Dependence of a Set of Functions
  8. Vector Norms and Inner Products

II. Matrices

  1. Product of Two Matrices
  2. The Operation of Transposition
  3. Inversion
  4. Inverses for Some special Matrices: Band Matrices
  5. The Row and Column spaces of a Matrix, Nullspace of a Matrix, Elementary Matrices


III  Generalized Inverses

  1. Right and Left Inverses
  2. Principle of Least Squares
  3. Matrix Methods for Least Squares Problems
  4. The Moore-Penrose Inverse

IV Linear Transformations

  1. Definition and Examples
  2. The Matrix of a Linear Transformation
  3. Transition Matrix and Change of Basis

V  The Eigenvalue Problem

  1. Eigenvalues and Eigenvectors
  2. Eigenvalues of Some Special Matrices
  3. Spectral Theorem
  4. Jordan 's Canonical Form
  5. Positive Definite Matrices

VI  Function of a Matrix

  1. Introduction
  2. Functions of a Matrix Defined by a Power Series
  3. Evaluation of exp( i A) by the Laplace Transform Method

VII  Irreducible and Monotone Matrices

  1. Irreducible Matrices
  2. Monotone Matrices
  3. Application to the Two-Point Linear Boundary Value Problem
  4. Nonlinear Two-Point Boundary Value Problem


Probability Models :

VIII  Introduction to Probability Theory

  1. Sample Space and Events
  2. Probabilities Defined on Events
  3. Conditional Probabilities
  4. Independent Events
  5. Bayes' Fomula

IX  Random variables

  1. Random Variables
  2. Discrete Random variables
  3. Continuous Random Variables
  4. Expectation of a random Variables
  5. Jointly Distributed Random variables
  6. Moment Generating Functions
  7. Limit Theorem
  8. Stochastic Processes

X  Conditional Probability and Conditional Expectation

  1. The Discrete case
  2. The Continuous case

XI. The Exponential Distribution and Poisson Process

  1. The Exponential Distribution
  2. The Poisson Process
  3. Generalizations of the Poisson Process

XII. Continuous-Time Markov Chains

  1. Continuous-time Markov Chains
  2. Birth and Death Processes
  3. Limiting Probabilities

Real and Complex Analysis :

XIII  Point-set topology

  1. Finite, countable, and uncountable sets
  2. Open and closed sets
  3. Metric spaces
  4. Compact sets
  5. Connected sets

XIV Numerical sequences and series

  1. Convergence of sequences
  2. Convergence of series

XV  Real-valued functions

  1. Continuous functions
  2. Intermediate value theorem
  3. Differentiability and differentiation
  4. Mean value theorem
  5. Taylor series
  6. Integrability and integration
  7. Fundamental theorem of calculus

XVI  Complex-valued functions

  1. Analytic functions
  2. Complex integration
  3. Cauchy integral theorem
  4. Residue theorem
  5. Conformal mapping

 

W. Rudin, Principles of Mathematical Analysis . McGraw-Hill, 1976.
E.B. Saff and A.D. Snider, Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics. Prentice-Hall, 2003.
Sheldon M. Ross, Introduction to Probability Models, Fifth Edition , Academic Press, Inc. 1993
Athanasios Papoulis, Probability, random Variables and Stochastic Processes, Second Edition, McGraw-Hill International Editions 1984
Riaz A. Usmani, Applied Linear Algebra , Marcel Dekker Inc., 1987
Iyengar Jain, Advanced Engineering Mathematics , Narosa Publishing House

 

The final grade will be computed from final exam (100%).