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Currently, I am an Associate Professor in the Department of Physics and Astronomy and an Adjunct Professor in the Department of Mathematics at the University of Manitoba. I am the Director Elect of the Winnipeg Institute for Theoretical Physics and an associate at the Center for Advanced Mathematical Sciences at the American University of Beirut.

 

Contact Information

Education

Professional Experience

Refereed Publications

Invited Talks

Contributed Talks

Seminars and Colloquia

Research Interest

Training of HQP

Teaching

Service

Honor Societies/Awards

Professional Societies

Languages

12th p-Adic Conference

Contact Information

Education

Professional Experience

Refereed Publications

                                                                                                                                                              I.                        Books

  1. Advances in Ultrametric Analysis, Proceedings of the 12th International Conference on p-Adic Functional Analysis, Khodr Shamseddine, editor, Contemporary Mathematics, American Mathematical Society, Volume 596, 2013.
  2. Advances in p-Adic and Non-Archimedean Analysis, Proceedings of the 10th International Conference on p-Adic and Non-Archimedean Analysis, Martin Berz and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 508, 2010, ISBN 978-0-8218-4740-4.

                                                                                                                                                       II.                        Papers

  1. A brief survey of the study of power series and analytic functions on the Levi-Civita fields, Khodr Shamseddine, Proceedings of the 12th International Conference on p-Adic Functional Analysis, Contemporary Mathematics, American Mathematical Society, Volume 596 (Advances in Ultrametric Analysis), 2013.
  2. One-variable and multi-variable calculus on a non-Archimedean field extension of the real numbers, Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 5 # 2, 2013, pp. 160-175.
  3. Characterization of compact and self-adjoint operators on Free Banach spaces of countable type over the complex Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, Journal of Mathematical Physics, Volume 54 # 2, 2013.
  4. New results on integration on the Levi-Civita field, Khodr Shamseddine, Indagationes Mathematicae, Volume 24 # 1, 2013, pp. 199-211.
  5. Preliminary notes on Fourier series for functions on the Levi-Civita field, Khodr Shamseddine and William Grafton, International Journal of Mathematical Analysis, Volume 6 # 19, 2012, pp. 941-950.
  6. On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers, Khodr Shamseddine and Todd Sierens, ISRN Mathematical Analysis, Volume 2012, Article ID 387053, 20 pages.
  7. Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field, Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 551 (Advances in Non-Archimedean Analysis), 2011, pp. 257-268.
  8. Nontrivial order preserving automorphisms of non-Archimedean fields, Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 547 (Function Spaces in Modern Analysis), 2011, pp. 217-225.
  9. On the topological structure of the Levi-Civita field, Khodr Shamseddine, Journal of Mathematical Analysis and Applications, Volume 368, 2010, pp. 281-292.
  10. Analysis on the Levi-Civita field, a brief overview, Khodr Shamseddine and Martin Berz, Contemporary Mathematics, American Mathematical Society, Volume 508 (Advances in p-Adic and Non-Archimedean Analysis), 2010, ISBN 978-0-8218-4740-4, pp. 215-237.
  11. The implicit function theorem in a non-Archimedean setting, Khodr Shamseddine, Trevor Rempel and Todd Sierens, Indagationes Mathematicae, Volume 20 # 4, 2009, pp. 603-617. [This paper is the result of my work with two NSERC-USRA students at the University of Manitoba in Summer 2009]
  12. Intermediate value theorem for analytic functions on a Levi-Civita field, Khodr Shamseddine and Martin Berz, Bulletin of the Belgian Mathematical Society-Simon Stevin, Volume 14, 2007, pp. 1001-1015.
  13. Generalized power series on a non-Archimedean field, Khodr Shamseddine and Martin Berz, Indagationes Mathematicae, Volume 17 # 3, 2006, pp. 457-477.
  14. Analytical properties of power series on Levi-Civita fields, Khodr Shamseddine and Martin Berz, Annales Mathématiques Blaise Pascal, Volume 12 # 2, 2005, pp. 309-329.
  15. Constrained second order optimization on non-Archimedean fields, Khodr Shamseddine and Vera Zeidan, Indagationes Mathematicae, Volume 14 # 1, 2003, pp. 81-101.
  16. Measure theory and integration on the Levi-Civita field, Khodr Shamseddine and Martin Berz, Contemporary Mathematics, American Mathematical Society, Volume 319 (Ultrametric Functional Analysis), 2003, ISBN: 0-8218-3320-0, pp. 369-387.
  17. Intermediate values and inverse functions on non-Archimedean fields, Khodr Shamseddine and Martin Berz, International Journal of Mathematics and Mathematical Sciences, Volume 30 # 3, 2002, pp. 165-176.
  18. On the existence and uniqueness of solutions of differential equations on the Levi-Civita field, Khodr Shamseddine, International Journal of Differential Equations and Applications, Volume 4 # 4, 2002, pp. 375-386.
  19. Infinite dimensionality of the  solution space of y’=0 on non-Archimedean fields, Khodr Shamseddine, International Journal of Differential Equations and Applications, Volume 4 # 1, 2002, pp. 25-30.
  20. One-dimensional optimization on  non-Archimedean fields, Khodr Shamseddine and Vera Zeidan, Journal of Nonlinear and Convex Analysis, Volume 2 # 3, 2001, pp. 351-361.
  21. Convergence on the Levi-Civita field and study of power series, Khodr Shamseddine and Martin Berz, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Proceedings of the Sixth International Conference on P-adic Analysis, July 2-9, 2000, ISBN 0-8247-0611-0, pp. 283-299.
  22. The differential algebraic structure of the Levi-Civita field and applications, Khodr Shamseddine and Martin Berz, International Journal of Applied Mathematics, Volume 3 # 4, 2000, pp. 449-464.
  23. Power series on the Levi-Civita field, Khodr Shamseddine and Martin Berz, International Journal of Applied Mathematics, Volume 2 # 8, 2000, pp. 931-952.
  24. Exception handling in derivative computation with non-Archimedean calculus, Khodr Shamseddine and Martin Berz, Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, A. Griewank, eds., SIAM, Philadelphia, Penn, 1996, pp. 37-51.
  25. COSY INFINITY and its applications to nonlinear dynamics, Martin Berz, Kyoko Makino, Khodr Shamseddine, Georg Hoffstatter and Weishi Wan, Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, A. Griewank , eds., SIAM, Philadelphia, Penn, 1996, pp. 363-367.

 

                                                                                                            III.                        Papers in Preparation

28.              B*-algebras of operators and study of positive operators on a free Banach space of countable type over the complex Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, to be submitted soon.

Invited Talks at Conferences

  1. Analysis on non-Archimedean ordered field extensions of the real numbers and applications, NUMTA2013 (Numerical Computations: Theory and Applications) International Conference and Summer School, Falerna, Italy, June 16-23, 2013
  2. B*-algebras of operators and study of positive operators on a free Banach space of countable type over the complex Levi-Civita field, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
  3. (Co-author, talk given by Jose Aguayo) Characterization of Compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
  4. (Co-author, talk given by Todd Sierens) On locally uniformly differentiable functions: the Inverse Function Theorem and the Implicit Function Theorem in a non-Archimedean setting, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
  5. Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field, 11th International Conference on p-Adic Functional Analysis, Université Blaise Pascal, Clermont-Ferrand, France, July 5-9, 2010.
  6. Analysis on a non-Archimedean field extension of the real numbers and applications, The Sixth Conference on Function Spaces, Southern Illinois University- Edwardsville, Illinois, USA, May 18-22, 2010.
  7. Analysis on the Levi-Civita field, a brief overview, Tenth International Conference on p-Adic and Non-Archimedean Analysis, Michigan State University, East Lansing, Michigan, USA, June 30- July 3, 2008.
  8. Intermediate value theorem for analytic functions on a Levi-Civita field, Ninth International Conference on P-adic Functional Analysis, University of Concepcion, Concepcion, Chile, July 10-14, 2006.
  9. Analytical properties of power series on Levi-Civita fields, Eighth International Conference on P-adic Functional Analysis, Clermont-Ferrand, France, July 5-9, 2004.
  10. Measure theory and integration on the Levi-Civita field, Seventh International Conference on P-adic Functional Analysis, Nijmegen, The Netherlands, June 17-21, 2002.
  11. The differential algebraic structure of the Levi-Civita field and applications, Ninth International Colloquium on Numerical Analysis and Computer Science with Applications, Plovdiv, Bulgaria, August 12-17, 2000.
  12. Convergence on the Levi-Civita field and study of power series, Sixth International Conference on P-adic Functional Analysis, Ioannina, Greece, July 2-9, 2000.
  13. Power series on the Levi-Civita field, Eighth International Colloquium on Numerical Analysis and Computer Science with Applications, Plovdiv, Bulgaria, August 13-18, 1999.

Contributed Talks at Conferences

  1. The Non-Archimedean Field R, Overview and Applications, International Conference on Scientific Computations, LAU, Beirut, Lebanon, March 1999.
  2. Non-Archimedean Structures as Differentiation Tools, Second LAAS International Conference on Computer Simulation, AUB, Beirut, Lebanon, September 1997.
  3. Exception Handling in Derivative Computation with Non-Archimedean Calculus, SIAM 96: Second International Workshop on Industrial and Applied Mathematics, Santa Fe, New Mexico, February 1996.

Invited Talks, Seminars and Colloquia at Universities

  1. Department of Physics and Astronomy and Winnipeg Institute for Theoretical Physics (joint colloquium), University of Manitoba, April 10, 2013.
  2. Science Department, Texas A & M University in Qatar, October 23, 2012.
  3. Department of Mathematics, Western Illinois University, August 31, 2012.
  4. Departments of Mathematics (Joint Mathematics Colloquium), Universidad del Bio-Bio and Universidad de Concepcion, Concepcion, Chile, December 5, 2011.
  5. Department of Mathematics, American University of Beirut, Beirut, Lebanon, July 21, 2011.
  6. Department of Mathematics, University of Wisconsin- Eau Claire, December 17, 2008.
  7. Winnipeg Institute of Theoretical Physics, November 5, 2008.
  8. Winnipeg Institute of Theoretical Physics, October 22, 2008.
  9. Department of Mathematics, University of Wisconsin- Eau Claire, July 7, 2007.
  10. Department of Mathematics, University of Wisconsin- Eau Claire, June 21, 2007.
  11. Department of Physics & Astronomy, University of Manitoba, November 24, 2006.
  12. Department of Mathematics, University of Manitoba, November 24, 2005.
  13. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, August 12, 2005.
  14. Center for Research in Applied Mathematics & Statistics (CRAMS), Business and Computer University College, Beirut, Lebanon, January 7, 2005.
  15. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, January 5, 2005.
  16. Department of Mathematics, Western Illinois University, November 11, 2004.
  17. Department of Mathematics, University of Wisconsin- Eau Claire, July 13, 2004.
  18. Department of Physics and Astronomy, Michigan State University, June 18, 2004.
  19. Department of Mathematics, University of Wisconsin- Eau Claire, April 30, 2004.
  20. Department of Mathematics, Western Illinois University, February 5, 2004.
  21. Department of Mathematics, Western Illinois University, February 14, 2003.
  22. Department of Mathematics and Computer Science, Lawrence Technological University, Michigan, May 14, 2002.
  23. Department of Mathematics, Michigan State University, October 8, 2001.
  24. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, July 19, 2000.
  25. National Superconducting Cyclotron Laboratory, Michigan State University, November 24, 1999.
  26. Department of Mathematics, American University of Beirut, Beirut, Lebanon, March 24, 1999.

Research Interests

My research interests and activities include various areas of non-Archimedean Analysis: power series and analytic functions, measure theory and integration, optimization, existence and uniqueness of solutions of differential equations, complex analysis, multivariable analysis and functional analysis. The focus of my research has been on the Levi-Civita fields which were first introduced by the Italian mathematician Tullio Levi-Civita at the end of the nineteenth century. Of those Levi-Civita fields, one (which we denote by R) is of particular interest; it is shown to be the smallest non-Archimedean field extension of the real numbers that is complete in the order topology and real closed. In fact, R is small enough so that the numbers of the field can be implemented on a computer; and this allows for many useful applications, one of which is the fast and accurate computation of the derivatives of real-valued functions up to high orders.

We studied two topologies on R: the valuation topology induced by the order on the field, and another weaker topology induced by a family of semi-norms, which we call weak topology. We showed that each of the two topologies results from a metric on R, that the valuation topology is not a vector topology while the weak topology is, and that R is complete in the valuation topology while it is not in the weak topology. Then we studied the properties of both topologies in detail; in particular, we gave simple characterizations of open, closed, and compact sets in both topologies. Finally, we showed that the metric which induces the weak topology is translation invariant.

We studied convergence of sequences and series in both topologies mentioned above, which led to an exhaustive study of power series. A handful of people had investigated power series on the Levi-Civita fields before, but all the previous studies had been restricted to the special case of power series with real coefficients. We have dropped that restriction and showed that power series on Levi-Civita fields have all the nice smoothness properties that real power series have. In particular, they satisfy the intermediate value theorem, the extreme value theorem, the mean value theorem and the inverse function theorem; they are infinitely often differentiable; and they are re-expandable around any point within their domain of convergence.

While it is a known fact that conventional continuity or differentiability are not sufficient to guarantee that a function on a closed interval of a non-Archimedean field be bounded or satisfy any of the common theorems of real calculus, we have shown that under mild conditions, differentiability is sufficient for the function to assume all intermediate values and a differentiable inverse function. We also showed that conventional differentiability is not the right one to study optimization questions on non-Archimedean fields in general; and based on a stronger concept of differentiability, we studied finite-dimensional optimization both with and without constraints. In both cases, we derived necessary and sufficient conditions of first and second order for a function to have a local minimum at a point of its domain.

We developed a measure theory and integration on the Levi-Civita field R. We introduced a measure that proved to be a natural generalization of the Lebesgue measure on the field of the real numbers and have similar properties. Then we introduced a family of simple functions from which we obtained a larger family of measurable functions and derived a simple characterization of such functions. We studied the properties of measurable functions, we showed how to integrate them over measurable sets, and we showed that the resulting integral satisfies similar properties to those of the Lebesgue integral of Real Analysis.

Most recently, together with my collaborators Jose Aguayo and Miguel Nova from Concepcion (Chile), we developed an operator theory on a Banach space over C:=R+iR. Let c_0 denote the space of all null sequences of elements of C. The natural inner product on c_0 induces the sup-norm of c_0. We showed that c_0 is not orthomodular; then we characterized those closed subspaces of c_0 with an orthonormal complement with respect to the inner product. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We presented a characterization of such normal projections as well as a characterization of other kinds of operators, the self-adjoint and compact operators on c_0. Then we worked on some B*-algebras of operators, including those mentioned above; we studied normal and Hilbert-Schmidt operators; and finally, we studied the properties of positive operators, which we then used to introduce a partial order on the B*-algebra of compact and self-adjoint operators on c_0 and studied the properties of that partial order.

Building on my previous research, I am establishing a research program in Non-Archimedean Analysis at the University of Manitoba. The long term aim of my research program is to develop a complete analysis which holds for any real closed Cauchy-complete non-Archimedean field extension F of the field of real numbers R whose skeleton group is Archimedean. The theorem of Hahn entails that any such field F is isomorphic to a subfield of a field of generalized power series with coefficients in R and exponents forming a well-ordered subset of some ordered abelian group G. As a part of my research program, I have designed several research projects, any one of which (or, in some cases, a combination thereof) could also be a suitable topic for an interested student's M.Sc. or Ph.D. thesis.

 

Training of Highly Qualified Personnel (at U of M)

        I.            Graduate Students

1.       William Grafton (September 2013- ); M.Sc. candidate (Mathematics).

2.       Angel Barria (January 2013- ); PhD candidate (Mathematics).

3.       Darren Flynn (July 2012- ); M.Sc. candidate (Physics).

4.       Andrew Senchuk (January 2010- ; I am co-advisor, with advisor: Prof. Gerald Gwinner); PhD candidate (Physics).

 

      II.            Undergraduate Students

o   Summer Students

1.       Gidon Bookatz (2013); NSERC-USRA student in the joint Math-Physics Honours program.

2.       William Grafton (2012); undergraduate summer student in the joint Math-Physics Honours program. William will start his M.Sc. in Mathematics under my supervision in September, 2013.

3.       James Roberts (2011); undergraduate summer Honours student, Department of Physics at the University of Winnipeg.

4.       William Grafton (2011).

5.       Todd Sierens (2010); undergraduate summer student in the joint Math-Physics Honours program; currently a PhD student at the University of Waterloo and the Perimeter Institute.

6.       Todd Sierens (2009); NSERC-USRA student in the joint Math-Physics Honours program.

7.       Trevor Rempel (2009); NSERC-USRA student in the joint Math-Physics Honours program; currently a PhD student at the University of Waterloo and the Perimeter Institute.

 

o   Undergraduate Honour Thesis:

8.       Cheng Tang (2012-2013).

 

Teaching

 

        I.            At the University of Manitoba (May 2008- present)

o   Summer 2013: MATH 8430, Non-Archimedean Operator Theory (a reading course)

 

 

 

 

 

 

 

 

 

 

  1. At Western Illinois University (August 2003- May 2008)

 

  1. At Michigan State University (January 2000- August 2003):

 

Service

 

        I.            At the University of Manitoba (May 2008- present)

 

a.      Graduate Students Advisory Committees

                                                              i.      May 2011- present: Mark McCrea; PhD in Physics.

 

b.      Thesis Internal Examiner

                                                              i.      2010: Bryan Penfound; M.Sc. in Mathematics.

                                                            ii.      2010: Chandra Podder; PhD in Mathematics.

                                                          iii.      2010: Mohammad Safi; PhD in Mathematics.

 

c.       Departmental Committees

                                                              i.      2012-2013:

1.      Honours/Majors Committee (Chair)

2.      Curriculum Committee (Chair)

3.      Term Work Appeals

                                                            ii.      2011-2012:

1.      Honours/Majors Committee (Chair)

2.      Curriculum Committee (Chair)

                                                          iii.      2010-2011:

1.      Honours/Majors Committee (Chair)

2.      Curriculum Committee

                                                          iv.      2009-2010:

1.      Honours/Majors Committee (Chair)

2.      Curriculum Committee

                                                            v.      2008-2009: Curriculum Committee

 

d.      Faculty of Science Committees

                                                              i.      2012-2013:

1.      Director Elect of the Winnipeg Institute for Theoretical Physics

2.      Committee on Courses and Programs (COCAP)

3.      The Local Discipline Committee

                                                            ii.      2011-2012:

1.      Committee on Courses and Programs (COCAP)

2.      The Local Discipline Committee

                                                          iii.      2010-2011: The Local Discipline Committee

                                                          iv.      2009-2010: The Local Discipline Committee

                                                            v.      2008-2009: The Local Discipline Committee

 

e.       International Committees and Professional Service

                                                              i.      Member of the Scientific Committee for the 13th International Conference on p-Adic Functional Analysis (to be held at University of Paderborn, Germany, August 2014).

                                                            ii.      Member of the Scientific Committee for NUMTA2013: Numerical Computations: Theory and Algorithms (Falerna, Italy, June 17-23, 2013)

                                                          iii.      Organizer and chair of the Scientific Committee for the 12th International Conference on p-Adic Functional Analysis (University of Manitoba, July 2-6, 2012)

                                                          iv.      Member of the Scientific Committee for the 11th International Conference on p-Adic Functional Analysis (Universite Blaise Pascal, France, July 5-9, 2010)

                                                            v.      Co-organizer and member of the Scientific Committee for the 10th International Conference on p-Adic and Non-Archimedean Analysis (Michigan State University, USA, June 30-July 3, 2008)

                                                          vi.      Reviewer for Zantrablatt Math (13 papers since 2009)

                                                        vii.      Referee of several papers for Math journals (6 papers since 2008.)

                                                      viii.      Referee of 4 research proposals (3 for Qatar Foundation and 1 for the Austrian Science Fund.)

 

f.       Community Service: Member of the Board of Directors for Campus Daycare since May 2011.

 

g.      Outreach: Volunteered at info days every year (2009-2013).

 

      II.            At Western Illinois University (August 2003- May 2008)

 

a.      Graduate Students Advisory Committees

                                                              i.      2003-2004: Javid Siddique; M.Sc. in Mathematics.

                                                            ii.      2003-2004: Duygu Inceoz; M.Sc. in Mathematics.

 

b.      Thesis External Examiner

                                                              i.      2008: Dodzi Attimu; PhD in Mathematics; Howard University (US).

 

c.       Departmental Committees

                                                              i.      2007-2008

1.      Graduate Committee (Chair)

2.      Personnel Committee

3.      Curriculum Committee

                                                            ii.      2006-2007

1.      Graduate Committee (Chair)

2.      Colloquium Committee

3.      Computer Affairs Committee

4.      Curriculum Committee

                                                          iii.      2005-2006

1.      Graduate Committee (Chair)

2.      Colloquium Committee

3.      Curriculum Committee

                                                          iv.      2004-2005

1.      Graduate Committee (Chair)

2.      Colloquium Committee

3.      Curriculum Committee

4.      Upper Division Curriculum Committee

                                                            v.      2003-2004

1.      High School Relations Committee

2.      Library Committee (Chair)

3.      Upper Division Curriculum Committee

 

As the Chair of the Graduate Committee (Fall 2004-Spring 2008), I led the efforts of that committee in recruiting and advising graduate students as well as assisting the Department Chair with various matters pertaining to the graduate program. I have led the efforts of the Graduate Committee in preparing the proposals for revisions in the graduate program including new structure, courses and tracks to better serve the needs of the incoming graduate students. I gathered ideas, thoughts and suggestions from the various groups in the Department about the structure and the contents of the courses in the revised program and designed a web page for that purpose to facilitate the discussions and the exchange of ideas in the Department. As a result, proposals for a revised program were submitted and approved by the Dean and the Graduate Council; and the revised program started in Fall 2006.

 

d.      College (Arts and Science) and University Committees

                                                              i.      2007-2008

1.      University Research Council

2.      Evidence of English Proficiency and Minimum TOEFL Requirements Committee

3.      College of Arts & Sciences Graduate Studies Committee

                                                            ii.      2006-2007

1.      University Research Council

2.      Evidence of English Proficiency and Minimum TOEFL Requirements Committee

3.      College of Arts & Sciences Graduate Studies Committee

                                                          iii.      2005-2006

1.      College of Arts & Sciences Graduate Studies Committee

 

   III.            At Michigan State University: Coordinator of the Math Learning Center (2002-2003), supervising all the tutors and teaching assistants who worked at the center.

 

Honor Societies and Awards

Professional Societies

Languages

Maintained by K. Shamseddine (last updated May 13, 2013). Top of home page.