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**Undergraduate Courses:**

Set theory, Probability , Statistics, Numerical analysis, Linear algebra, Discrete mathematics , Mathematical analysis, Operations research, Differential Geometry, Topology, Advanced Numerical Analysis, Measure theory

**Graduate Courses:**

Research topic, Advanced numerical analysis, The finite element method, Numerical linear algebra, Distributions Theory, Evolution Equations, Spectral theory, Measure and Integration, Advanced Analysis and Applications.

- MATH 315 - Advanced Numerical Analysis
- MATH 317 - Computational Methods
- MATH 344 - Real Analysis II
- MATH 653 - Measure and Integration
- MATH 654 - Advanced Analysis and Applications
- MATH 753 - Spectral Theory
- MATH 312 - Operation Research

**Advanced Numerical Analysis - MATH 315**

**Lecture Meeting: **Tuesday 0800 - 1000

**Catalog Description**

Numerical Methods for boundary value problems: shooting, parallel shooting and finite difference methods for linear and nonlinear problems. Finite difference methods for partial differential equations, derivation and error analysis, consistency, stability and convergence. Numerical methods for the matrix eigenvalue problems: power method and its variants, Householder method, the QR algorithm. Iterative methods for solving linear systems: Jacobi, Gauss-Seidel and SOR methods, derivation and error analysis. Numerical Methods for initial value problems: Euler, Taylor, Runge-Kutta, multistep, predictor-corrector methods.

**Course Objectives**

The course aims to provide students with the specialist knowledge in advanced Numerical Analysis. With this overall aim, the course strives to enable students to: Understand analytical, developmental and technical principles that relate to Numerical Linear Algebra, Numerical Methods for solving Differential Equations, and Numerical Optimization, develop the academic abilities required to solve problems and applications in Numerical Analysis and/or Numerical Optimization and critically assess relevant aspects of the industry, and demonstrate an ability to initiate and sustain in-depth research in Numerical Analysis or Numerical Optimization.

**Course Outcomes**

1) Knowledge and Understanding: During the lecture the student understands the nature and operations of Numerical Analysis, demonstrates familiarity with theories and concepts used in Numerical Analysis, and identifies the steps required to carry out a piece of research on a topic in Numerical Analysis.

3) Intellectual Skills: By the end of the course the student is expected to recognize and apply appropriate theories, principles and concepts relevant to Numerical Analysis, critically assess and evaluate the literature within the field of Numerical Analysis, analyze and interpret information from a variety of sources relevant to Numerical Analysis.

4) Practical Skills: By the end of the course student will have the ability to compare the computational methods for advantages and drawbacks, choose the suitable computational method among several existing methods, implement the computational methods using any of existing programming languages, testing such methods and compare between them, identify the suitable computational technique for a specific type of problems, and develop the computational method that is suitable for the underlying problem. 3) Transferable Skills: Within the lectures the student is able to transfer ideas and experience, work effectively as a part of a group and independently.

Student Outcomes

K1-Knowledge of basic theorems and concepts in the different areas of mathematics.

K2-Knowledgeof the implementation of theories in problem solving.

K3-Knowledge of different areas of research in mathematics.

I1-Ability to understand the different math concepts and be able to implement them in our everyday problems.

P1-Efficient use of computers, laboratories and soft wares to handle problems that are difficult to be solved

manually.

P2-Understanding of professional and ethical responsibilities

P3-Effecient use of the techniques, skills, and tools of modern mathematics

Download course syllabus as PDF

**Computational Methods - MATH 317**

**Lecture Meeting: Wednesday **08:00 - 10:00

Catalog Description

Fundamental methods of computational methods and applications; numerical algorithms, linear algebra, differential equations; computer simulation; vectorization, parallelism, optimization and examples on scientific applications.

Course Objectives

Teach the students various computational methods that enable them solving model problems in different fields of Mathematics. Teach the students to use available computational packages such as MATLAB in solving problems from different fields of Mathematics and Mathematical Statistics.

**Course Outcomes**

1) Knowledge and Understanding: During the lecture the student learns to model different types of problems and select the suitable package and existing programs in these packages to solve the given problems

2) Intellectual Skills: By the end of the course the student is expected to solve real-life problems reflecting the student ability to identify the suitable computational technique for a specific type of problems.

3) Practical Skills: By the end of the course student will have the ability to compare the computational methods for advantages and disadvantages, choose the suitable computational method among several existing methods and implement the computational methods using existing computational software.

4) Transferable Skills: Within the lectures the student is able to transfer ideas and experience, work effectively as a part of a group and independently.

**Student Outcomes**

K1-Knowledge of basic theorems and concepts in the different areas of mathematics.

I1-Ability to understand the different math concepts and be able to implement them in our everyday problems.

I2-Ability to consider problems that could be solved by implementing concepts from different areas in

mathematics

P3- Efficient use of the techniques, skills, and tools of modern mathematics

S1-Functioning in multi-disciplinary teams

Download course syllabus as PDF

**Real Analysis II - MATH 344**

**Lecture Meeting: **Wednesday 1000 - 1130 **
**Tuesday 1200 - 1330

Catalog Description

Riemann integral, convergence of sequences and series of functions, functions of several variables, limit of integral of a sequence of functions, contraction principle.

This course aims to provide students with the specialist knowledge necessary for basic concepts in Real Analysis. More precisely, it strives to enable students to learn basic concepts about functions of bounded variation, grasp basic concepts about the total variation, learn about Riemann-Stieltjes integrals , sequences and series of functions.

1) Knowledge and Understanding: Learn the theory of Riemann-Stieltjes integrals, to be aquainted with the ideas of the total variation and to be able to deal with functions of bounded variation.

2) Intellectual Skills: Develop a reasoned argument in handling problems about functions, especially those that are of bounded variation.

3) General and Transferable Skills: Develop the ability to reflect on problems that are quite significant in the field of real analysis.

K1-Knowledge of basic theorems and concepts in the different areas of mathematics.

K2-Knowledge of the implementation of theories in problem solving.

I1-Ability to understand the different math concepts and be able to implement them in our everyday problems.

I2-Ability to consider problems that could be solved by implementing concepts from different areas in

mathematics.

I3-Ability to identify, formulate, and solve problems.

P2-Understanding of professional and ethical responsibilities

S2-Communicate ideas effectively in graphical, oral, and written media

Download course syllabus as PDF

**Measure and Integration - MATH 653**

**Lecture Meeting: **Monday 16:00 - 19:00

**Catalog Description**

General properties of measures, construction of Lebesgue measure, measurable sets & measurable functions, Lebesgue integration, convergence theorems, Lp spaces, product measures, Radon-Nikodym theorem, decomposition and differentiation of measures.

**Course Objectives**

Teach the students the theory of Operators and some application in Partial differential Equations and related subjects.

**Course Outcomes**

1) Knowledge and Understanding: During the lecture the student learns to model different types of problems and select the suitable package and existing programs in these packages to solve the given problems

2) Intellectual Skills: By the end of the course the student is expected to solve real-life problems reflecting the student ability to identify the suitable computational technique for a specific type of problems.

3) Practical Skills: By the end of the course student will have the ability to compare the computational methods for advantages and disadvantages, choose the suitable computational method among several existing methods and implement the computational methods using existing computational software.

4) Transferable Skills: Within the lectures the student is able to transfer ideas and experience, work effectively as a part of a group and independently.

Download course syllabus as PDF

**Advanced Analysis and Applications - MATH 654**

**Lecture Meeting: **Thursday 1600 - 1900

**Catalog Description**

Banach and Hilbert spaces, linear operators, Riez representation theorem, Hahn Banach theorem , uniform boundedness theorem, open mapping theorem, inverse mapping theorem, closed graph theorem, dual spaces, introduction to spectral theory, operators on normed spaces, Sobolev spaces and applications to linear differential equations.

**Course Objectives**

Teach the students the theory of Hilbert spaces, Sobolev spaces and some application in Partial differential Equations and related subjects.

**Course Outcomes**

1) Knowledge and Understanding: During the lecture the student learns to model different types of problems and select the suitable package and existing programs in these packages to solve the given problems.

2) Intellectual Skills: By the end of the course the student is expected to solve real-life problems reflecting the student ability to identify the suitable computational technique for a specific type of problems.

3) Practical Skills: By the end of the course student will have the ability to compare the computational methods for advantages and disadvantages, choose the suitable computational method among several existing methods and implement the computational methods using existing computational software.

4) Transferable Skills: Within the lectures the student is able to transfer ideas and experience, work effectively as a part of a group and independently.

Download course syllabus as PDF

**Special Theory - MATH 753**

**Lecture Meeting: **Friday 16:00 - 19:00

**Catalog Description**

Hilbert spaces, spectrum of a bounded operator, symbolic calculus: bounded and unbounded self-adjoint operator, L∞ spectral theorem, L2 spectral theorem, Stone’s theorem.

**Course Objectives**

Teach the students the theory of Operators and some application in Partial differential Equations and related subjects.

**Course Outcomes**

1) Knowledge and Understanding: During the lecture the student learns to model different types of problems and select the suitable package and existing programs in these packages to solve the given problems

2) Intellectual Skills: By the end of the course the student is expected to solve real-life problems reflecting the student ability to identify the suitable computational technique for a specific type of problems.

3) Practical Skills: By the end of the course student will have the ability to compare the computational methods for advantages and disadvantages, choose the suitable computational method among several existing methods and implement the computational methods using existing computational software.

4) Transferable Skills: Within the lectures the student is able to transfer ideas and experience, work effectively as a part of a group and independently.

Download course syllabus as PDF

**Operation Research - MATH 312**

**Lecture Meeting: Monday **11:00 - 13:00

Catalog Description:

Modeling linear programming and deriving methods for solving them using algorithms such as:geometrical method, simplex method, dual simplex method, transportation algorithms. Problems without constraint will be also discussed, where various numerical methods apply.

**Course Objectives:**

This course provides students with the specialist knowledge necessary for basic concepts in linear programming . It enables them to learn the fundamental aspects of optimization problems, basic concepts about geometrical, simplex, dual simplex methods and numerical methods for unconstraint problems.

**Course Outcomes:**

1) Knowledge and Understanding: The student learns linear program problems, be familiar with theories and concepts used throughout the course and deals with different applications related to applied mathematics.

2) Intellectual Skills: Learn the fundamental aspects of numerical algorithms in order to solve optimization problems and implement it in a lot of examples.

3) General and Transferable Skills: Develop the ability to reflect on problems that are quite significant in the field of mathematics and work effectively both in a team and independently.

Student Outcomes:

K1-Knowledge of basic theorems and concepts in the different areas of mathematics.

I2-Ability to consider problems that could be solved by implementing concepts from different areas in

mathematics

P1-Efficient use of computers, laboratories and softwares to handle problems that are difficult to be solved

manually.

P2-Understanding of professional and ethical responsibilities

P3- Efficient use of the techniques, skills, and tools of modern mathematics

S3-Recognize and respond to the need for lifelong and self learning for a successful career

Download course syllabus as PDF

Inverse Problems: Studying the D-bar method : Establishing error estimates for the regularisation and effect of the geometry.

Establishing stability estimates for regularisation.

Semigroups: Theoretical approximation for the Dirichlet-to-Neumann semigroup by using Chernoff procedure. Numerical scheme for solving elliptic and parabolic linear problem with dynamical boundary conditions.

Establishing stability estimates for regularisation.

Semigroups: Theoretical approximation for the Dirichlet-to-Neumann semigroup by using Chernoff procedure. Numerical scheme for solving elliptic and parabolic linear problem with dynamical boundary conditions.

1. T. El Arwadi Méthode De Dbar Pour La Résolution Du Problème De Conductivité, Nouvelle Approche ( Europian University Edition)

2. T. El Arwadi Error estimates for reconstructed conductivities via the D-bar method. Numerical Functional Analysis and Optimization ,Vol. 33, Iss. 1, 2012 ,pp 21-38

3. M.A. Cherif and T. El Arwadi . D-Bar Method for Smooth and Nonsmooth Conductivities .Vol. 35, Iss. 2, 2014 ,pp 198-222

4. M.A. Cherif, T. El Arwadi, H. Emamirad, J.M Sac Épée The Dirichlet-to-Neumann semigroup acts as a magnifying glass Semigroup Forum, Vol. 88, Iss. 3, 2014 ,pp 753-767

5. R. Ahmad, T. El Arwadi, H. Chrayteh, J.M Sac Épée A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup J. App. Math, Vol. 2015, Article ID 429641, http : dx.doi.org/10.1155/2015/429641

6. T. El Arwadi, S. Dib, T. Sayah A Priori and A Posteriori error estimates for an elleptic problem with dynamical boundary conditions App. Math. Inf. Sc., 6 (9), DOI :10.12785/amis/100121 (2015) pp 2205-2217

7. T. El Arwadi, A. Zaart A Novel 5x5 Edge Detection Operator for Blood Vessel Images British Journal of Applied Science and Technology 11(3) :1-10, 2015, Article no.BJAST.19967 ISSN : 2231-0843

8. A. Zaart, T. El Arwadi A New Edge Detection Method for CT-Scan Lung Images Journal Of Biomedical Engineering And Medical Imaging 2(5), ISSN :2055-1266, DOI : 10.14738/jbemi.25.1453

9. T. El Arwadi, T. Sayyah Lp estimates for Dirichlet-to-Neumann operator and applications EJDE, Vol. 2015, No. 258 pp 1-8, ISSN : 1072-6691

10. R. Ahmad, T. El Arwadi, H. Chrayteh, J.M Sac Épée A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions J. Math. Res., Vol 8, No 2 (2016), http ://dx.doi.org/10.5539/jmr.v8n2p1

11. S. Al Kontar, T. El Arwadi, H. Chrayteh, J.M Sac Épée Stability of the D-bar reconstruction method for complex conductivities The Australian Journal of Mathematical Analysis and Applications Vol. 13, No 1, pp :1-14 (2016)

12. T. El Arwadi, V. Flammang, G.Rhin, J.M Sac Épée Extension of the notion of Mahler measure to a certain class of holomorphic functions. Properties and applications. Results in Math 72, no. 1-2, 787-791 (2017)

13. S. Al Kontar, T. El Arwadi, H. Chrayteh, J.M Sac Épée About the D-Bar reconstruction method for complex conductivities : Error estimates (To appear)

14. T. El Arwadi, A. Wehbe, W. Youssef Theoretical and Numerical observability of the Bresse Beam (Submitted)

2. T. El Arwadi Error estimates for reconstructed conductivities via the D-bar method. Numerical Functional Analysis and Optimization ,Vol. 33, Iss. 1, 2012 ,pp 21-38

3. M.A. Cherif and T. El Arwadi . D-Bar Method for Smooth and Nonsmooth Conductivities .Vol. 35, Iss. 2, 2014 ,pp 198-222

4. M.A. Cherif, T. El Arwadi, H. Emamirad, J.M Sac Épée The Dirichlet-to-Neumann semigroup acts as a magnifying glass Semigroup Forum, Vol. 88, Iss. 3, 2014 ,pp 753-767

5. R. Ahmad, T. El Arwadi, H. Chrayteh, J.M Sac Épée A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup J. App. Math, Vol. 2015, Article ID 429641, http : dx.doi.org/10.1155/2015/429641

6. T. El Arwadi, S. Dib, T. Sayah A Priori and A Posteriori error estimates for an elleptic problem with dynamical boundary conditions App. Math. Inf. Sc., 6 (9), DOI :10.12785/amis/100121 (2015) pp 2205-2217

7. T. El Arwadi, A. Zaart A Novel 5x5 Edge Detection Operator for Blood Vessel Images British Journal of Applied Science and Technology 11(3) :1-10, 2015, Article no.BJAST.19967 ISSN : 2231-0843

8. A. Zaart, T. El Arwadi A New Edge Detection Method for CT-Scan Lung Images Journal Of Biomedical Engineering And Medical Imaging 2(5), ISSN :2055-1266, DOI : 10.14738/jbemi.25.1453

9. T. El Arwadi, T. Sayyah Lp estimates for Dirichlet-to-Neumann operator and applications EJDE, Vol. 2015, No. 258 pp 1-8, ISSN : 1072-6691

10. R. Ahmad, T. El Arwadi, H. Chrayteh, J.M Sac Épée A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions J. Math. Res., Vol 8, No 2 (2016), http ://dx.doi.org/10.5539/jmr.v8n2p1

11. S. Al Kontar, T. El Arwadi, H. Chrayteh, J.M Sac Épée Stability of the D-bar reconstruction method for complex conductivities The Australian Journal of Mathematical Analysis and Applications Vol. 13, No 1, pp :1-14 (2016)

12. T. El Arwadi, V. Flammang, G.Rhin, J.M Sac Épée Extension of the notion of Mahler measure to a certain class of holomorphic functions. Properties and applications. Results in Math 72, no. 1-2, 787-791 (2017)

13. S. Al Kontar, T. El Arwadi, H. Chrayteh, J.M Sac Épée About the D-Bar reconstruction method for complex conductivities : Error estimates (To appear)

14. T. El Arwadi, A. Wehbe, W. Youssef Theoretical and Numerical observability of the Bresse Beam (Submitted)

- I organized a workshop on Partial differential equations 2015 (Day of Partial Differential Equations).

- 2012 - LSMS annual meeting (member of the scientific committee).

- 2013 - LSMS annual meeting (member of the organizing committee).

- 2014 - LSMS annual meeting (member of the organizing committee).

- 2014 - LAAS annual Workshop (member of the scientific committee).

- 2015- LSMS annual meeting (member of the organizing committee).

- 2015 - LAAS annual Workshop (member of the scientific committee).

- 2016- LSMS annual meeting (member of the organizing committee).

- 2016 - LAAS annual Workshop (member of the scientific committee).

Recent Oral Presentations

- 12/2012 LSMS meeting New approach of D-bar methods

- 07/2012 9th AIMS workshop (Orlando) New approach of D-bar methods

- 12/2012 Lebanese University - Fanar New approach of D-bar methods

- 09/2013 Saint-Joseph University On the Dirichlet-to-Neumann semigroup

- 07/2015 Saint-Joseph University The electrical impedance tomography and inverse problems

- 10/2016 Université Badji Mokhtar Annaba New approach of D-bar method for EIT and inverse convection problem

- 03/2016 Lebanese University – Hadath Electrical impedance tomography and D-bar method.

- Curriculum committee, member.

- Comprehensive exam committee, member

- Students Activities committee, member.

- Graduate seminar in Mathematics and Computer sciences, organizer

- 2012 -2015 Lebanese Society for the Mathematical Sciences (National Relations)

- 2015- Lebanese Society for the Mathematical Sciences (International Relations)

International Collaborators

- A. Tamasan (UCF, Florida, USA)

- M.A Cherif (University of Sfax, Tunisia)

- M. Uesaka (University of Tokyo, Japan)

- H. Emamirad (University of Poitiers, France)

- J.M Sac-Epée (Lorraine university, France)

- 2012 - LSMS annual meeting (member of the scientific committee).

- 2013 - LSMS annual meeting (member of the organizing committee).

- 2014 - LSMS annual meeting (member of the organizing committee).

- 2014 - LAAS annual Workshop (member of the scientific committee).

- 2015- LSMS annual meeting (member of the organizing committee).

- 2015 - LAAS annual Workshop (member of the scientific committee).

- 2016- LSMS annual meeting (member of the organizing committee).

- 2016 - LAAS annual Workshop (member of the scientific committee).

Recent Oral Presentations

- 12/2012 LSMS meeting New approach of D-bar methods

- 07/2012 9th AIMS workshop (Orlando) New approach of D-bar methods

- 12/2012 Lebanese University - Fanar New approach of D-bar methods

- 09/2013 Saint-Joseph University On the Dirichlet-to-Neumann semigroup

- 07/2015 Saint-Joseph University The electrical impedance tomography and inverse problems

- 10/2016 Université Badji Mokhtar Annaba New approach of D-bar method for EIT and inverse convection problem

- 03/2016 Lebanese University – Hadath Electrical impedance tomography and D-bar method.

- Curriculum committee, member.

- Comprehensive exam committee, member

- Students Activities committee, member.

- Graduate seminar in Mathematics and Computer sciences, organizer

- 2012 -2015 Lebanese Society for the Mathematical Sciences (National Relations)

- 2015- Lebanese Society for the Mathematical Sciences (International Relations)

International Collaborators

- A. Tamasan (UCF, Florida, USA)

- M.A Cherif (University of Sfax, Tunisia)

- M. Uesaka (University of Tokyo, Japan)

- H. Emamirad (University of Poitiers, France)

- J.M Sac-Epée (Lorraine university, France)