Courses 8

 

College:

Science

Department:

Mathematics

Course Title:

Discreet Mathematics

Course Code:

MATH 462

Credit Hours:

3

Prerequisite:

Basics of Mathematics Math 251

Text Books:

1.

Introduction to Integral Equations with Applications, 2nd  ed. Jerri, (1999), Wiely-Interscience.

2.

3.

Course Description:

Basic of discrete mathematics- Formal logic -Theorem sets. Algebraic basic rules- semigroups- Identities-Sets-Linear code. Graph theory: Graphs- Directed graphs-Paths- Walks - Eulerian and Hamiltonian graphs- Shortest  path - Planar graphs –Path colouring – Trees- Spanning trees –Different algorithms.

Viability of the account :Table of transition- Planning graphs- Homomorphisms, Isomorphism -The partial regression functions- Languages and rules- The regular expressions.

Learning Objectives:

-       To provide the student with knowledge of logical thinking

-       To provide the student with the basic concepts discrete mathematics

-       To teach student how to apply software on these topics

Grading: 

No.

Assessment

Evaluation

1.

Mid Term Exam 1

25

2.

Mid Term Exam 2

25

3.

Homework

5

4.

Quiz

5

5.

Final Exam

40

Total

100 %

Methods of Teaching: (Lectures, Laboratory, Individual exercises, In-class discussion, Selection of Readings,…)

1.

Lectures

2.

Individual exercises

3.

In-class discussion

4.

Course Outline:

Week

Lecture Topics

1.

Basic of discrete mathematics

2.

Formal logic -Theorem sets.

3.

Algebraic basic rules- semigroups

4.

Identities-Sets-Linear code.

5.

Graph theory: Graphs- Directed graphs-Paths- Walks -

6.

Eulerian and Hamiltonian graphs

7.

Shortest  path - Planar graphs –Path colouring

8.

Trees

9.

Spanning trees

10.

Different algorithms.

11.

Viability of the account :Table of transition- Planning graphs-

12.

Homomorphisms

13.

Isomorphism

14.

The partial regression functions- Languages and rules-

15.

The regular expressions.

College:

Faculty of science

Department:

Mathematics

Course Title:

Differential Geometry

Course Code:

Math 463

Credit Hours:

3

Prerequisite:

Math 204, Math 305

Text Books:

1.

P. Petersen, Riemannian Geometry, Springer, New York, 1998.

2.

Geodesics: Christoffel symbols, "straight lines," more geodesics

3.

Course Description:

Study the Geometry of Curves and Surfaces in three dimensional space using calculus techniques. Topics include Curves: arclength, tangent vector, curvaturebinormal vector, torsion Gauss Curvature: normal section, principal curvature Surfaces in E3: surfaces of revolution, parallelsFirst Fundamental Form: metric form, intrinsic propertySecond Fundamental Form: Frenet Frame, normal curvature Gauss Curvature in Detail: principal curvature. .

Learning Objectives:

Students who are successful in this course will improve in the following general education areas:

 - differential geometry (with an emphasis on curvature),

-  Surfaces in E3

- Geodesics: Christoffel symbols. We will spend about half of our time on the theories of curves and surfaces in E3.

Grading: 

No.

Assessment

Evaluation

1.

Mid Term Exam 1

25%

2.

Mid Term Exam 2

25%

3.

Homework

5%

4.

Quiz

5%

5.

Final Exam

40%

Total

100 %

Methods of Teaching: (Lectures, Laboratory, Individual exercises, In-class discussion, Selection of Readings,…)

1.

Lectures

2.

Tutorials

3.

Homework

4.

Quizzes

Course Outline:

Week

Lecture Topics

1.

Curves:  Arclength, Tangent vector, curvature.

2.

Curvature, binormal vector

3.

Torsion, Gauss Curvature

4.

normal section, principal curvature

5.

Theories of curves

6.

Surfaces in E3: surfaces of revolution, parallels

7.

First Term Exam  ,  

8.

Theories of surfaces

9.

First Fundamental Form: metric form, intrinsic property

10.

Second Fundamental Form: Frenet Frame, normal curvature

11.

Gauss Curvature in Detail: principal curvature

12.

Second Term Exam  ,  

13.

Introduction to Christoffel symbols

14.

Revision

15.

College:

Science

Department:

Mathematics

Course Title:

Functional Analysis

Course Code:

MATH 462

Credit Hours:

3

Prerequisite:

Linear Algebra 1: MATH241, Real Analysis 1 (Math311)

Text Books:

1.

Lax. P. ''Functional analysis'', Wiley- inter science, 2002.

2.

Akhiezer, N. I. and Glazman, I. M. '' Theory of linear operators in Hilbert space'' Frederrich Ugar publishing Co. New York Vol I (1961), Vol II (1963).

3.

 Kolmogorov A.N., Fomin S.V. ''Elements of the theory of functions and functional analysis'' Dover publications, (1999)

Course Description:

Vector spaces:linear subspace - linear dependence and linear combination- dimension and basis- spanning- direct sum decomposion for V(F). Inner product spaces:Cauchy Schwarz  inequality- Minkowski inequality- polarization identity – orthogonality and orthogonal sets of vectors - orthogonalization (Gram-Schmidt). Normed spaces and metric spaces:Holder inequality- general Minkowski inequality- metric and metric spaces- some topological notion in metric spaces- convergent and completeness – Cauchy sequence – complete metric –continuity and uniform continuity on metric spaces- contraction mappings. Banach spaces and Hilbert spaces:Linear manifold – orthogonal system – Fourier coefficient. Linear operators:bounded operators –continuous – linear functional in Hilbert H- adjoint operator- self adjoint operator – normal operator- unitary and isometric operators – projection operators- closed operator- graph of an operator- eigenvalues and eigenvector – symmetric operators- positive operators- the formal differential operator ( - spectrum of self adjoint operator.

Learning Objectives:

-       To allow the student study the theoretical spaces.

-       To allow the student acquire some properties of sequences that are defined on the theoretical spaces.

Grading: 

No.

Assessment

Evaluation

1.

Mid Term Exam 1

25

2.

Mid Term Exam 2

25

3.

Homework

5

4.

Quiz

5

5.

Final Exam

40

Total

100 %

Methods of Teaching: (Lectures, Laboratory, Individual exercises, In-class discussion, Selection of Readings,…)

1.

Lectures

2.

Individual exercises

3.

In-class discussion

4.

Course Outline:

Week

Lecture Topics

1.

Vector spaces: linear subspace - linear dependence and linear combination- dimension and basis

2.

Spanning- direct sum decomposion for V(F).

3.

Inner product spaces: Cauchy Schwarz  inequality- Minkowski inequality- polarization identity.

4.

Orthogonality and orthogonal sets of vectors - orthogonalization (Gram-Schmidt).

5.

Normed spaces and metric spaces: Holder inequality- general Minkowski inequality.

6.

Metric and metric spaces- some topological notion in metric spaces.

7.

Convergent and completeness – Cauchy sequence – complete metric.

8.

Continuity and uniform continuity on metric spaces- contraction mappings.

9.

Banach spaces and Hilbert spaces: Linear manifold – orthogonal system – Fourier coefficient.

10.

Linear operators: bounded operators –continuous – linear functional in Hilbert H.

11.

Adjoint operator- self adjoint operator – normal operator- unitary and isometric operators.

12.

Projection operators- closed operator- graph of an operator.

13.

Eigenvalues and Eigenvector.

14.

Symmetric operators- positive operators.

15.

The formal differential operator- spectrum of self adjoint operator.