## Course Content

Analysis I (MAT 111)

Real Numbers; The Field of Real Numbers, Ordering and The Completeness Axioms, Interior Point,

Accumulation Point. The Concept of a Function; Types and Properties of Functions, The Concept of

Countabilitiy, Infinite and Countable Infinite Sets; The Concept of Supremum and Infimum. Sequences;

Convergence and Divergence of Sequences, Convergence Theorems. Elemanter Functions; Logaritmic and

Exponential Functions, Hyperbolic Functions. Series; Convergence of Series, Absolute Convergence,

Convergence Theorems. Power Series, Radius of Convergence. Trigonometric Functions, Limits of

Functions.

Analytic Geometry I (MAT 121)

Vector Space, Subspace and Spaning Space, Concepts of Linear Dependence, Bases and Dimension,

Vectors in Affine Spaces and Operations with Vectors, Barycentric Dependence. Operations on Vectors in

Euclidean Space. Line in a Plane and Basic Problems. Plane in a Three Dimensional Space and Basic

Problems. Parabola and Central Conics, Ellipse, Hyhperbola.

Linear Algebra I (MAT 131)

Definitions of Group, Ring and Field. Systems of Linear Equations. Matrices; Elemanter Row

Operations, Matrix Multiplication, Invertible Matrices. Vector Spaces; Subspaces, Basis, Dimension,

Coordinates. Linear Transformations; Algebra of Linear Transformations, Isomorphism, Matrix

Representations of Linear Transformations. Linear Functionals. Inverses of Linear Transformations.

Determinants; The Determinant Function, Properties of Determinant, Sarus’s Rule, Cramer’s Rule

Computer 1 (MAT 141 )

Basic Use of Computer; Ms Dos, Windows, Word.

Analysis II (MAT 112)

Continuous Functions, The Derivative. Rolle’s Theorem, The Mean Value Theorem, Cauchy Mean

Value Theorem. Applications of Derivatives, Taylor and Maclaurin Series. Power Series Expansion of

Functions . Integral; Definition and Properties, Integrable Functions. Techniques of Integration.

Analytic Geometry II (MAT 122)

Circle and Analytic Investigation of Circle, General Quadratic Equation , Line in Three Dimensional

Space and Basic Problems, Surfaces, Sphere, Cone and Cylinder, Surfaces of Revolution, Quadratic Surfaces

of Canonical Equations, Change of Variables in Three Dimensional Space and General Quadratic Equation.

Linear Algebra II (MAT 132)

Polynomials; Algebra of Polynomials, Divisibility in the Ring of Polynomials, Ideals. Eigenvalue

Equations; Eigenvalue Polynomials of Similar Matrices, Diagonilazations, Dimension of Eigenvalue Space,

Minimal Polynomial, Canonical Forms. Invariant. Inner Product Spaces; Standard Inner Product, Norm.

Quadratic Form, Orthogonality, Orthogonal Set, Orthonormal Set, Orthogonal Basis, Orthonormal Basis,

Orthogonal Projection. Bessel’s Inequality.

Computer 2 (MAT 142)

Excel, Powerpoint, Internet.

Analysis III (MAT 211)

Cartesian Spaces, Sequences in Rn and Convergence, Uniform Convergence, Uniform Convergence of

Sequences and Series of Functions, Fourier Series of a Functions with respect to Orthonormal Systems,

Fourier Series, Improper Integrals.

Algebra I (MAT 221)

Concept of a Set, Construction of Number Sets. Definition of Congruence and Basic Properties.

Complete and Reduced Systems, Euler’s ϕ–Function. Polynomial Congruences Two variables Linear

Congruences, Quadratic Congruences, Order of an Integer with respect to Modulo-m, Primitive Roots,

Index, Quadratic Reciprocity Theorem, Multivariables Linear Congruence Systems.

Differential Equations I (MAT 231)

Basic Definitions. Existence and Uniqueness Theorems of Solutions of Differential Equations,

First Order Equations and Methods of Solutions, Higher Order Differential Equations.

Differential Geometry I (MAT 331)

Euclidean Spaces, Tangent Vectors, Directional Derivatives, Curves in E3, Differential Forms, Theory

of Curves, Frénet Formulas, Covariant Derivatives, Field of Frames, Connection Forms. Structure Equations.

Matrix Theory (MAT 224)

Gauss Elimination and Inverse Substitution. Nonsingular Systems of Initial Basic Submatrices.

Methods of Factorizations. Lower, Upper Envelops and Band Matrices, Matrix Norms, Iterative Methods.

Inverses of Matrices and Determinants, Determination of Eigenvalues and Eigenvectors.

Computer 3 (MAT 241)

Pascal Programming Language.

Analysis IV (MAT 212)

Limit and Continuity of Functions of Two Variables. Partial Derivatives, Chain Rule, Differential,

Exact Differential, Directional Derivatives. Maxima and Minima of Functions of Two Variables, Lagrange

Multipliers (Constrained Extremum). Implicit Functions, Implicit Mapping Theorem, Inverse Functions.

Curves in Plane and Space, Line Integrals. Multiple Integrals and Applications.

Algebra II (MAT 222)

Group Theory, Permutation Groups, Subgroups, Cyclic Groups. Complexes of a Group and Calculus

with Complexes, Group Homomorphisms and Isomorphisms.

Differential Equations II (MAT 232)

Series Solutions of Differential Equations, Systems of Equations, Laplace Transforms.

Differential Geometry II (MAT 332)

Surfaces in E3, Differential Forms on Surfaces, Shape Operator, Normal Curvature, Gaussian

Curvature, Special Curves of a Surface, Fundamental Equations, Forms.

Computer 4 (MAT 242)

Pascal Programming Language.

Theory of Complex Functions I ( MAT 311)

Complex Numbers, Analytic Functions, Elemanter Functions, Complex Integration, Local Properties

of Analytic Functions. Residue Theorem and Applications.

Algebra III (MAT 321)

Rings, Integral Domain, Ideals and Quotient Rings. Homomorphism of Rings, Polynomial Rings,

Divisibility in an Integral Domain, Euclidean Domains and Field Extensions.

Partial Differential Equations (MAT 361)

First Order Homogeneous Linear Differential Equations, First Order Nonhomogeneous Linear

Differential Equations,, Pfaff’s Forms, Methods of Solutions of Nonlinear Partial Differential Equations,

Second Order Partial Differential Equations and Methods of Solutions.

Numerical Analysis I (MAT 415)

Numerical Error Sources, Error Analysis for Linear Systems, Classifications of Numbers, Methods for

Findings Fixed Points, Methods for Findings the Roots of Functions of One Variable, Order of Convergence

and Methods for Accelerating Convergence, Methods of Solutions for Nonlinear Systems.

Computer 5 (MAT 341)

Java Programming Language.

Numerical Analysis ( MAT 318)

Numerical Integration and Differentiation. Initial Value Problems for Ordinary Differenrial Equations.

Boundary Value Problems for Ordinary Differenrial Equations.

Theory of Complex Functions II (MAT 312).

Representation Theorems, Partial Fractions, Mittag-Leffler Theorem, Infinite Products, Canonical

Representation, Weierstrass Theorem. Special Functions; Gamma, Beta and Gauss Functions. Mappings;

Conformality. Fractional Linear Mappings, Cross Ratio and Symmetry. Basic Mapping Problems. Mapping

of a Polygon, Schwarz - Christoffel Formula, Elliptic Integrals, Complex Fourier Series.

Numerical Analysis II (MAT 416)

Weierstrass Theorem, Techniques of Polynomial Interpolation, Techniques of Other Interpolation The

Least Squares Method, Interpolation for Systems of Two Variables.

Probability (MAT 362 )

Probability. Sample Space, Sample Point and Events, Counting Rules of Sample Points. Permutations,

Combinations. Binomial Theorem. Probability of an Event, Probability Axioms, Some Probability Rules,

Geometric Probability, Conditional Probability, Independent Events, Bayes’s Theorem. Random Variables;

Discrete and Continuous Random Variables, Two Dimensional Random Variables, Expected Value,

Variance and Properties. Moments. Chebysev’s Inequality. Some Special Discrete Distributions; Bernoulli,

Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson and Uniform Distributions.

Computer 6 (MAT 342)

Java Programming Language.

Topology I ( MAT 413)

Sets, Operations on Sets, Indexed Sets, Relations, Equivalence Relations. Functions; Definition of a

Function, Inverse Functions, Inverse Images, Composition of Functions. Topological Spaces; Definition of

Topology Closed Sets, Topologies Defined by Functions, Interior, Exterior and Boundary of a Set

Accumulation Points, Bases, Subbases and Products; Bases, Finite Products of Topological Spaces,

Subbases, Genaral Product Spaces. Continuous Functions; Definition of a Continuous Function, Open

Functions and Homeomorphisms. Identification Topology, Quotient Spaces. Compactness, Compactness in

Rn .

Functional Analysis (MAT 411)

Classical Inequalities, Metric Spaces, Normed Linear Spaces, Banach Spaces Bounded Linear

Operators, Linear Functionals and Hahn-Banach Theorem, Baire Category Theorem and Consequences;

Uniform Boundedness Principle, Open Mapping Theorem, Inverse Mapping Theorem and Closed Graph

Theorem.

Statistics (MAT 451 )

Normal Distribution, Standard Normal Distribution, Normal Approximation to Binomial Distribution.

Moments Exerted Functions of Normal Distribution. Uniform, Exponential, Gamma and Beta Distributions.

Relations Betwen Distribution. Concept of Sample, Sample Selection. Organization of Data; Frequency

Distribution, Graphical Representations, Measures of Central Tendency, Measures of Variability. Sampling

Mean and Properties of Variance, Point Estimation, Interval Estimation. Test of Hypothesis for Mass

Parameters.

Applied Mathematics I (MAT 445)

Selected Topics in Applied Mathematics

Computer 7 (MAT 441)

Mathematica Programming Language.

Real Analysis (MAT 412)

Algebras of Sets, Functions of a Real Variable, Lebesque Measure, Lebesque Integral, Classical

Banach Spaces.

Topology II ( MAT 414)

Separation and Countability Axioms; Separation Axioms; Hausdorff Spaces, Regular and Normal

Spaces, The First Countability Axiom, The Second Countability Axiom. Convergence; Sequences,

Convergence in the First Countable Spaces, Definitions of Net and Filter. Connected and Compact Spaces;

Connected Spaces and Properties, Components and Locally Connected Spaces, Compactness and the

Properties of Compact Spaces, Compactness in Rn , Varieties of Compactness.

Mathematical Statistics (MAT 452)

Goodness of Fit, Chi - Square Test, Goodness of Fit with Binomial Distribution, Goodness of Fit with

Poisson Distribution, Tests of Independence, Regression Analysis, Model and Estimation of Parameters,

Confidence Intervals and Testing of Hypothesis, Mutivariables Linear Regression, Test of Linearity of

Regression, Correlations, Variance Analysis.

Applied Mathematics II (MAT 446)

Fredholm and Volterra Integral Equations. Relations of Integral Equations to Initial Value and

Boundary Value Problems and Green’s Function

Computer 8 (MAT 442)

Mathematica Programming Language.

Calculus I (MMAT 151)

Functions, Transcendental Functions, Limit and Continuity, Derivative, Applications of Derivative,

Integration, Techniques of Integration.

General Mathematics (BMAT 151)

Numbers, Functions, Graphs of Trigonometric and Exponential Functions. Permutation, Combination,

Binomial Formula and Probability Vectors, Vectors, Determinants and Systems of Linear Equation.

Calculus II (MMAT 152)

Applications of Integration, Improper Integrals, Conic Sections, Polar Coordinates, Vectors and

Analytic Geometry in Space, Functions of Several Variables and Partial Derivatives, Multiple Integrals.

Biostatistics (BMAT 152)

Statistics and Graphs, Frequency Distributions, Means, Variances, Probability, Distribution Theory,