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Currently, I am a tenured (full) professor in the Department of Physics and Astronomy and an adjunct professor in the Department of Mathematics at the University of Manitoba. I am also a former Ian C. P. Smith Integrated Science Faculty Scholar (July1, 2017- December 31, 2021), a former Associate Head for Undergraduate Studies in the Department of Physics and Astronomy (July 1, 2018- June 30, 2021), a former Director of the Winnipeg Institute for Theoretical Physics (January 1, 2015- December 31, 2016), and an associate at the Center for Advanced Mathematical Sciences at the American University of Beirut. Prior to arriving at the University of Manitoba, I was a tenured associate professor in the Department of Mathematics at Western Illinois University.

 

Contact Information

Education

Refereed Publications

Invited Talks

Contributed Talks

Seminars and Colloquia

Research Interest

Training of HQP

Conference Organization

Teaching

Service

Honor Societies/Awards

Professional Societies

Languages

Research Group Photo

2018 Merit Award in Teaching, Service and Research

Contact Information

Education

Refereed Publications

                                                               I.                  Books

  1. Advances in Ultrametric Analysis, Proceedings of the 14th International Conference on p-Adic Functional Analysis, Alain Escassut, Cristina Perez-Garcia, and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 704, 2018, ISBN: 978-1-4704-3491-5.
  2. Advances in Non-Archimedean Analysis, Proceedings of the 13th International Conference on p-Adic Functional Analysis, Helge Glockner, Alain Escassut and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 665, 2016, ISBN: 978-1-4704-1988-2.
  3. 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, ISBN: 978-0-8218-9142-1.
  4. 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.      On a Lebesgue-like measure on the Levi-Civita space ℛ j, Mateo Restrepo Borrero and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 16 # 4, 2024, pp. 303-326.

2.      On the Complex Levi-Civita field: Algebraic and Topological Structures, and Foundations for Analysis, Khodr Shamseddine, Khayyam Journal of Mathematics, Volume 10 # 1, 2024, pp. 70-89.

3.      On a new measure on the Levi-Civita field ℜ, Mateo Restrepo Borrero, Vatsal Srivastava, and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 15 # 1, 2023, pp. 1-22.

4.      A weaker smoothness criterion for the inverse function theorem, intermediate value theorem, and mean value theorem in a non-Archimedean setting, Khodr Shamseddine and Aaron Shalev, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 14, Suppl. # 1, 2022, pp. S45-S58.

5.      On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field, Khodr Shamseddine, Proceedings of the Edinburgh Mathematical Society, Volume 65, 2022, pp. 691-704.

6.      On non-Archimedean valued fields: a survey of algebraic, topological and metric structures, analysis and applications, Khodr Shamseddine and Angel Barria Comicheo, Advances in Non-Archimedean Analysis and Applications - The p-adic Methodology, a special volume in STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, 2021, pp. 209-254.

7.      Taylor's theorem, the inverse function theorem and the implicit function theorem for weakly locally uniformly differentiable functions on non-Archimedean spaces, Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 13 # 2, 2021, pp. 148-165.

8.      On computational applications of the Levi-Civita field, Darren Flynn and Khodr Shamseddine, Journal of Computational and Applied Mathematics, Volume 382, 2021.

9.      On the topological structure of the Hahn field and convergence of power series, Darren Flynn and Khodr Shamseddine, Indagationes Mathematicae, Volume 30 # 5, 2019, pp. 773-795.

10.  On an operator theory on a Banach space of countable type over a Hahn field, Khodr Shamseddine and Changying Ding, Proceedings of the 11th ISAAC Congress (Växjö, Sweden, August 2017), Analysis, Probability, Applications, and Computation, April 2019, pp. 267-282.

11.  Summary on non-Archimedean valued fields, Angel Barria Comicheo and Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 704 (Advances in Ultrametric Analysis), 2018, pp. 1-36.

12.  Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem, Gidon Bookatz and Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 704 (Advances in Ultrametric Analysis), 2018, pp. 49-67.

13.  On Integrable Delta Functions on the Levi-Civita Field, Darren Flynn and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 10 # 1, 2018, pp. 32-56.

14.  Positive operators on a free Banach space over the Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 9 # 2, 2017, pp. 122-137.

15.  A local mean value theorem for functions on non-Archimedean field extensions of the real numbers, Khodr Shamseddine and Gidon Bookatz, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 8 # 2, 2016, pp. 160-175.

16.  Measure theory and Lebesgue-like integration in two and three dimensions over the Levi-Civita field, Khodr Shamseddine and Darren Flynn, Proceedings of the 13th International Conference on p-Adic Functional Analysis, Contemporary Mathematics, American Mathematical Society, Volume 665 (Advances in Non-Archimedean Analysis), 2016, pp. 289- 325.

17.  On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field, Alpar Meszaros and Khodr Shamseddine, Journal of Contemporary Mathematical Analysis, Volume 50 # 2, 2015, pp. 53-62.

18.  Analysis on the Levi-Civita field and computational applications, Khodr Shamseddine, Applied Mathematics and Computation, Volume # 255, 2015, pp. 44-57.

19.  Inner product on B*-algebras of operators on a free Banach space over the Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, Indagationes Mathematicae, Volume 26 # 1, 2015, pp. 191-205.

20.  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, pp. 269-280.

21.  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.

22.  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.

23.  New results on integration on the Levi-Civita field, Khodr Shamseddine, Indagationes Mathematicae, Volume 24 # 1, 2013, pp. 199-211.

24.  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.

25.  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.

26.  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.

27.  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.

28.  On the topological structure of the Levi-Civita field, Khodr Shamseddine, Journal of Mathematical Analysis and Applications, Volume 368, 2010, pp. 281-292.

29.  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.

30.  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]

31.  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.

32.  Generalized power series on a non-Archimedean field, Khodr Shamseddine and Martin Berz, Indagationes Mathematicae, Volume 17 # 3, 2006, pp. 457-477.

33.  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.

34.  Constrained second order optimization on non-Archimedean fields, Khodr Shamseddine and Vera Zeidan, Indagationes Mathematicae, Volume 14 # 1, 2003, pp. 81-101.

35.  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.

36.  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.

37.  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.

38.  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.

39.  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.

40.  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.

41.  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.

42.  Power series on the Levi-Civita field, Khodr Shamseddine and Martin Berz, International Journal of Applied Mathematics, Volume 2 # 8, 2000, pp. 931-952.

43.  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.

44.  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.

Invited Talks at Conferences

1.     On the complex Levi-Civita field: algebraic and topological structures, and foundations for analysis, 15th International Conference on p-adic Analysis and Dynamical Systems, an online meeting, July 11 and 14, 2023.

2.     Introduction to Non-Archimedean Analysis (mini-course: 3 hours), the Canadian Mathematical Society Summer Meeting, St. John’s, Newfoundland, June 3-6, 2022.

3.     On non-Archimedean Valued Fields and Ultrametric Spaces: the Hahn Fields and Levi-Civita Fields, CIMPA-CIMAT Research School, p-Adic Numbers, Ultrametric Analysis, and Applications, Guanajuato, Mexico, May 23-31, 2022.

4.     On non-Archimedean valued fields: a survey of algebraic, topological and metric structures, analysis and applications, Eighth International Conference on p-adic Mathematical Physics and its Applications, an online conference, May 17-28, 2021.

5.     Calculus theorems for locally uniformly differentiable functions on a non-Archimedean ordered field extension of the real numbers, Seventh International Conference on p-adic Mathematical Physics and its Applications, Covilha, Portugal, September 30-October 4, 2019.

6.     One-variable and Multi-variable Integral Calculus over the Levi-Civita Field and Applications, Sixth International Conference on p-adic Mathematical Physics and its Applications, Mexico City, Mexico, October 23-27, 2017.

7.     On the Levi-Civita Fields: Introduction and Summary of Selected Recent Research, 11th Congress of the International Society for Analysis, its Applications and Computations (ISAAC), Växjö, Sweden, August 14-18, 2017.

8.     Calculus on a non-Archimedean field extension of the real numbers: The intermediate value theorem, mean value theorem, inverse function theorem and implicit function theorem, 14th International Conference on p-Adic Functional Analysis, Aurillac, France, June 30-July 5, 2016.

9.     On the Levi-Civita fields: introduction and survey of recent research, 14th International Conference on p-Adic Functional Analysis, Aurillac, France, June 30-July 5, 2016.

10.  One-variable and Multi-variable Integral Calculus over the Levi-Civita Field and Applications, NUMTA2016 (Numerical Computations: Theory and Applications), The 2nd International Conference and Summer School, Calabria, Italy, June 19-25, 2016.

11.  Characterization of compact and self-adjoint operators, and study of positive operators on a Banach space over a non-Archimedean field, International Conference on p-Adic Mathematical Physics and its Aplications, Belgrade, Serbia, September 7-12, 2015.

12.  New results on the Lebesgue-like measure and integration theory on the Levi-Civita field and applications, 13th International Conference on p-Adic Functional Analysis, Paderborn, Germany, August 12-16, 2014.

13.  On positive operators on a Banach space over the complex Levi-Civita field, The Seventh Conference on Function Spaces, Southern Illinois University- Edwardsville, Illinois, USA, May 20-24, 2014.

14.  Preliminaries in non-Archimedean Functional Analysis, The Seventh Conference on Function Spaces, Southern Illinois University- Edwardsville, Illinois, USA, May 20-24, 2014.

15.  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.

16.  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.

17.  (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.

18.  (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.

19.  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.

20.  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.

21.  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.

22.  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.

23.  Analytical properties of power series on Levi-Civita fields, Eighth International Conference on P-adic Functional Analysis, Clermont-Ferrand, France, July 5-9, 2004.

24.  Measure theory and integration on the Levi-Civita field, Seventh International Conference on P-adic Functional Analysis, Nijmegen, The Netherlands, June 17-21, 2002.

25.  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.

26.  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.

27.  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. Elements of an operator theory on the space c0 over a non-Archimedean valued field, 46th Canadian Operator Symposium, University of Manitoba, June 4-8, 2018.
  2. The Non-Archimedean Field R, Overview and Applications, International Conference on Scientific Computations, LAU, Beirut, Lebanon, March 1999.
  3. Non-Archimedean Structures as Differentiation Tools, Second LAAS International Conference on Computer Simulation, AUB, Beirut, Lebanon, September 1997.
  4. 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 Mathematics and Statistics, Universidad de la Frontera, Temuco, Chile, May 20, 2022.
  2. Department of Physics and Engineering Physics, University of Saskatchewan, March 9, 2021.
  3. Department of Physics and Astronomy and Winnipeg Institute for Theoretical Physics (joint colloquium), University of Manitoba, March 8, 2019.
  4. Department of Mathematics, Universidad de Concepcion, Concepcion, Chile, March 23, 2018.
  5. Department of Mathematics, American University of Beirut, Beirut, Lebanon, August 31, 2017.
  6. Numerical Calculus Laboratory, University of Calabria, Rende, Italy, June 28, 2016.
  7. Department of Mathematics, American University of Beirut, Beirut, Lebanon, July 22, 2015.
  8. Department of Mathematics (Functional Analysis seminar, part II), University of Manitoba, March 17, 2015.
  9. Department of Mathematics (Functional Analysis seminar, part I), University of Manitoba, March 10, 2015.
  10. Department of Mathematics, University of Manitoba, March 21, 2014.
  11. Department of Physics, University of Regina, March 7, 2014.
  12. Department of Physics and Engineering Physics, University of Saskatchewan, March 6, 2014.
  13. Department of Mathematics & Statistics (Algebra seminar), University of Saskatchewan, March 6, 2014.
  14. Department of Physics and Astronomy and Winnipeg Institute for Theoretical Physics (joint colloquium), University of Manitoba, April 10, 2013.
  15. Science Department, Texas A & M University in Qatar, October 23, 2012.
  16. Department of Mathematics, Western Illinois University, August 31, 2012.
  17. Departments of Mathematics (Joint Mathematics Colloquium), Universidad del Bio-Bio and Universidad de Concepcion, Concepcion, Chile, December 5, 2011.
  18. Department of Mathematics, American University of Beirut, Beirut, Lebanon, July 21, 2011.
  19. Department of Mathematics, University of Wisconsin- Eau Claire, December 17, 2008.
  20. Winnipeg Institute of Theoretical Physics, November 5, 2008.
  21. Winnipeg Institute of Theoretical Physics, October 22, 2008.
  22. Department of Mathematics, University of Wisconsin- Eau Claire, July 7, 2007.
  23. Department of Mathematics, University of Wisconsin- Eau Claire, June 21, 2007.
  24. Department of Physics & Astronomy, University of Manitoba, November 24, 2006.
  25. Department of Mathematics, University of Manitoba, November 24, 2005.
  26. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, August 12, 2005.
  27. Center for Research in Applied Mathematics & Statistics (CRAMS), Business and Computer University College, Beirut, Lebanon, January 7, 2005.
  28. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, January 5, 2005.
  29. Department of Mathematics, Western Illinois University, November 11, 2004.
  30. Department of Mathematics, University of Wisconsin- Eau Claire, July 13, 2004.
  31. Department of Physics and Astronomy, Michigan State University, June 18, 2004.
  32. Department of Mathematics, University of Wisconsin- Eau Claire, April 30, 2004.
  33. Department of Mathematics, Western Illinois University, February 5, 2004.
  34. Department of Mathematics, Western Illinois University, February 14, 2003.
  35. Department of Mathematics and Computer Science, Lawrence Technological University, Michigan, May 14, 2002.
  36. Department of Mathematics, Michigan State University, October 8, 2001.
  37. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, July 19, 2000.
  38. National Superconducting Cyclotron Laboratory, Michigan State University, November 24, 1999.
  39. 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: one-variable and multi-variable calculus, power series and analytic functions, measure theory and integration, optimization, existence and uniqueness of solutions of differential equations, complex analysis, and functional analysis over non-Archimedean valued fields. 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 ordered field extension of the real numbers that is complete in the topology induced by the order 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 have studied in my research group 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 dropped that restriction and showed that power series on the 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 ordered 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. We generalized the results to two and three dimensions. In particular, we defined a Lebesgue-like measure on R2 (resp. R3). Then we defined measurable functions on measurable sets using analytic functions in two (resp. three) variables and showed how to integrate those measurable functions using iterated integration. The resulting double (resp. triple) integral satisfies similar properties to those of the single integral as well as those properties satisfied by the double and triple integrals of real calculus.

Together with my collaborators Jose Aguayo and Miguel Nova from Concepcion (Chile), we developed an operator theory on the Banach space c0 over C:=R+iR, where c0 denotes the space of all null sequences of elements of C. The natural inner product on c0 induces the sup-norm of c0. We showed that c0 is not orthomodular; then we characterized those closed subspaces of c0 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 c0. 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 c0 and studied the properties of that partial order.

                                                                                                                                       

While the Levi-Civita field R is interesting to study in detail for the reasons stated above, I have also expanded my research focus to include any non-Archimedean field extension of the real numbers that is real closed and complete in the topology induced by the order and whose Hahn group is Archimedean; such a field is denoted by F. For example, we studied the properties of weakly locally uniformly differentiable functions at a point or on an open subset of F or Fn and we proved local versions of the intermediate value theorem, the mean value theorem and Taylor's theorem for weakly locally uniformly differentiable functions on F. We also proved the inverse function theorem and implicit function theorem for weakly locally uniformly differentiable functions from Fn to Fn and from Fn to Fm (m < n), respectively. Moreover, the work on the topological structure as well as on the integration theory and its applications on the Levi-Civita field R has recently been extended to the field F.

 

 

 

Training of Highly Qualified Personnel (at U of M)

      I.         Postdoctoral Fellows:

1.     Angel Barria Comicheo (March 1-August 31, 2020); Department of Mathematics and Department of Physics and Astronomy (joint appointment).

2.     Angel Barria Comicheo (March 1-August 31, 2019); Department of Mathematics.

   II.         Graduate Students

1.     Arij Alameh (September 2021- ); candidate for M.Sc. in Physics.

2.     Darren Flynn (September 2014- August 2019); PhD in Physics.

3.     Angel Barria Comicheo (January 2013- October 2018); PhD in Mathematics.

4.     Andrew Senchuk (January 2011- May 2018; I was co-advisor, with advisor: Prof. Gerald Gwinner); PhD in Physics.

5.     William Grafton (September 2013- August 2016); M.Sc. in Mathematics.

6.     Darren Flynn (July 2012- August 2014); M.Sc. in Physics.

 

 III.         Undergraduate Students

o   Summer Students

1.     Mateo Restrepo Borrero (2023); undergraduate summer student, Department of Mathematics, Universidad Nacional de Colombia, Colombia.

2.     Aaron Shalev (2023); undergraduate summer student in the joint Math-Physics Honours program.

3.     Mateo Restrepo Borrero (2022); MITACS Globalink intern (from Colombia).

4.     Vatsal Srivastava (2022); MITACS Globalink intern (from India).

5.     Aaron Shalev (2022); NSERC-USRA student in the joint Math-Physics Honours program.

6.     Jeremy Croitor (2021); NSERC indigenous -USRA student in the joint Math-Physics Honours program.

7.     Jeremy Croitor (2020); NSERC indigenous -USRA student in the joint Math-Physics Honours program.

8.     Yudong Chen (2019); MITACS Globalink intern (from China).

9.     Zhenghang Du (2019); MITACS Globalink intern (from China).

10.  Changying Ding (2017); MITACS Globalink intern (from China).

11.  Jeremy Friesen (2017); NSERC-USRA student in the joint Math-Physics Honours program.

12.  Gidon Bookatz (2016); undergraduate summer student in the joint Math-Physics Honours program.

13.  Ryan Sherbo (2015); NSERC-USRA student in the joint Math-Physics Honours program.

14.  Gidon Bookatz (2015); NSERC-USRA student in the joint Math-Physics Honours program.

15.  Gidon Bookatz (2014); NSERC-USRA student in the joint Math-Physics Honours program.

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

17.  William Grafton (2012); undergraduate summer student in the joint Math-Physics Honours program.

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

19.  William Grafton (2011).

20.  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.

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

22.  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:

21.  Cheng Tang (2012-2013).

 

Teaching

 

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

 

o      Winter 2025: PHYS 7590, Electromagnetic Theory

 

o      Fall 2024:

Ø  PHYS 3496, Mathematical Physics II

Ø  PHYS 1020, General Physics I

 

o      Winter 2024: PHYS 7590, Electromagnetic Theory

 

o      Fall 2023:

Ø  PHYS 3496, Mathematical Physics II

Ø  PHYS 1020, General Physics I

 

o      Winter 2023:

Ø  PHYS 7590, Electromagnetic Theory

Ø  PHYS 3496, Mathematical Physics II

 

o      Fall 2022: No teaching

 

o      Winter 2022: on research leave

 

o      Fall 2021:

 

Ø  PHYS 3496, Mathematical Physics II

Ø  PHYS 7440, Foundations of Non-Archimedean Analysis and Applications in Physics (a reading course)

 

o      Fall 2020: PHYS 3496, Mathematical Physics II

 

o      Fall 2019: PHYS 3496, Mathematical Physics II

 

o      Winter 2019: PHYS 2496, Mathematical Physics I

 

o      Fall 2018: PHYS 3496, Mathematical Physics II

 

o      Winter 2018: on research leave

 

o      Fall 2017: Teaching credit as an Ian C. P. Smith Integrated Science (SIS) Faculty Scholar.

 

o      Winter 2017: PHYS 2490, Theoretical Physics II

 

o      Fall 2016:

Ø  PHYS 2390, Theoretical Physics I

Ø  PHYS 3650, Classical Mechanics II

 

o      Winter 2016: PHYS 2490, Theoretical Physics II

 

o      Fall 2015:

Ø  PHYS 1020, General Physics I

Ø  PHYS 2390, Theoretical Physics I

 

o      Winter 2015: PHYS 2490, Theoretical Physics II

 

o      Fall 2014:

Ø  PHYS 1020, General Physics I

Ø  PHYS 2390, Theoretical Physics I

 

o      Winter 2014: PHYS 2490, Theoretical Physics II

 

o      Fall 2013:

Ø  PHYS 1020, General Physics I

Ø  PHYS 2390, Theoretical Physics I

 

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

 

o      Winter 2013:

Ø  MATH 8430, Non-Archimedean Analysis (a reading course)

Ø  PHYS 2490, Theoretical Physics II

Ø  PHYS 3640, Electro- and Magnetodynamics and Special Relativity

 

o      Fall 2012: PHYS 2390, Theoretical Physics I

 

o      Winter 2012:

Ø  PHYS 2490, Theoretical Physics II

Ø  PHYS 3640, Electro- and Magnetodynamics and Special Relativity

 

o      Fall 2011: on sabbatical leave

 

o      Winter 2011:

Ø  PHYS 2490, Theoretical Physics II

Ø  PHYS 3640, Electro- and Magnetodynamics and Special Relativity

 

o      Fall 2010: PHYS 7590, Electromagnetic Theory

 

o      Winter 2010:

Ø  PHYS 2490, Theoretical Physics II

Ø  PHYS 3640, Electro- and Magnetodynamics and Special Relativity

 

o      Fall 2009: PHYS 7590, Electromagnetic Theory

 

o      Winter 2009:

Ø  PHYS 2490, Theoretical Physics II

Ø  PHYS 3640, Electro- and Magnetodynamics and Special Relativity

 

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

o      Spring 2008

Ø  MATH 133, Calculus and Analytic Geometry

Ø  MATH 391, Writing in the Mathematical Sciences

o      Fall 2007

Ø  MATH 137, Applied Calculus I

Ø  MATH 651, Elements of Modern Analysis

o      Spring 2007

Ø  MATH 133, Calculus and Analytic Geometry

Ø  MATH 391, Writing in the Mathematical Sciences

o      Fall 2006

Ø  MATH 137, Applied Calculus I

Ø  MATH 551, Methods of Classical Analysis

o      Spring 2006

Ø  MATH 133, Calculus and Analytic Geometry

Ø  MATH 391, Writing in the Mathematical Sciences

o      Fall 2005

Ø  MATH 137, Applied Calculus I

Ø  MATH 599, Special Topics: Methods of Classical Analysis

o      Spring 2005

Ø  MATH 137, Applied Calculus I

o      Fall 2004

Ø  MATH 137, Applied Calculus I

Ø  MATH 430, Multivariable Calculus

o      Spring 2004

Ø  MATH 137, Applied Calculus I

Ø  MATH 435, Introduction to Real Variables

o      Fall 2003

Ø  MATH 137, Applied Calculus I

Ø  MATH 311, Linear Algebra

 

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

o      2002-2003: MATH 234, Multivariable Calculus

o      2001-2002: MATH 234, Multivariable Calculus

o      Spring 2000, Fall 2000 and Spring 2001: MATH 235, Differential Equations.

 

Service

 

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

 

a.     PhD Students Advisory Committees

1.     Iris Halilovic, PhD in Physics: May 2023- present

2.     Brock Klippenstein, PhD in Physics: April 2023- present

3.     Naz Roshanshah, PhD in Physics: May 2021- present

4.     Brett Meggison, PhD in Physics: April 2021- present

5.     Erica Franzmann, PhD in Astrophysics: August 2015- September 2022 (successfully defended her thesis)

6.     Brad Cownden, PhD in Physics: November 2015- July 2020 (successfully defended his thesis)

7.     L.J. Zhou, PhD in Physics: May 2016- December 2018 (successfully defended his thesis)

8.     Ievgen Bilokopytov, PhD in Mathematics: November 2014- December 2018 (successfully defended his thesis)

9.     Mark McCrea, PhD in Physics: May 2011- January 2017 (successfully defended his thesis)

10.  Chandra Podder; PhD in Mathematics: Successfully defended his thesis in 2010

11.  Mohammad Safi; PhD in Mathematics: Successfully defended his thesis in 2010.

 

b.     M.Sc. Students Examining Committees

1.     Brett Meggison, M.Sc. in Physics: 2020

2.     Mitul Patel, M.Sc. in Physics: 2019

3.     Mahnaz Alavinejad, M.Sc. in Mathematics: 2016

4.     Jesse Bellec, M.Sc. in Physics: 2015

5.     Lindsay Simpson, M.Sc. in Mathematics: 2015

6.     Mahdi Kamaee, M.Sc. in Chemistry: 2014

7.     Fereshteh Nazari, M.Sc. in Mathematics: 2014

8.     Bryan Penfound, M.Sc. in Mathematics: 2010.

 

c.     Departmental Committees

1.     Outreach and Recruitment Committee (Member): Fall 2022-

2.     Undergraduate Studies Committee (member): Fall 2021-

3.     Undergraduate Studies Committee (Chair): Fall 2019- June 2021

4.     Curriculum Committee: 2008-2011 (Member); 2011-2019 (Chair)

5.     Management Committee (member): Fall 2019- June 2021

6.     Budget Committee (Member): 2018-2019

7.     Nominating Committee: 2014-2015 and 2018-2019

8.     Term Work Appeals (Chair): 2012-2019

9.     Honours/Majors Committee: 2009-2015 (Chair); 2017- 2019 (Member)

 

As the Associate head for Undergraduate Studies, I assisted the Department Head with all undergraduate matters; for example, I chaired the Undergraduate Studies Committee (see below), I decided on undergraduate students awards and I reviewed applications for course credit transfers.

 

As the chair of the Undergraduate Studies Committee, formerly the Curriculum Committee, (2011- 2021), I worked with other members of the Committee on making necessary curriculum and course changes. Then I presented those changes to COCAP for approval before being forwarded to the University Senate. Our undergraduate programs underwent an external review in 2015, which led to an overhaul of all honours and major programs:

·       I actively participated in preparing the report that was distributed to the undergraduate program review committee prior to their visit as well as in meetings with the review committee during the visit (March 2-4, 2015) and in drafting the Department’s response to the review committee.

·       Following the recommendations in the report from the undergraduate review committee, I led the efforts in 2015-2017 of the Curriculum Committee and the Department on making major changes to our undergraduate programs as well as the joint honours programs with Chemistry, Computer Science and Mathematics. Then I presented the package of changes for approval by COCAP (Faculty of Science Committee on Courses and Programs), then by the Faculty Council and finally by SCCCC (the Senate Committee on Curriculum and Course Changes). The changes were approved by all of the aforementioned committees and they became effective as of the fall 2018 term.

Moreover, as the chair of the Curriculum Committee, I coordinated the preparation at U of M of the 2015 CAP University Prize exam (which was written throughout Canada on Feb 3, 2015); and then I coordinated the grading of all written exams.

 

As the chair of the Honours/Majors Committee (2009-2015), I assigned a mentor for each of the students in our undergraduate programs and I served as a mentor myself for all the students in the Physics & Math Joint Honours program. Moreover, I assisted the Department Head in selecting the qualified students for the undergraduate awards.

 

d.     Faculty of Science Committees

1.     The Faculty-based Promotion Committee: 2020-2023

2.     The Local Discipline Committee: 2008- 2021

3.     Committee on Students, Programs and Undergraduate Degrees (SPUD): July 2018- June 2021

4.     Committee on Courses and Programs (COCAP): 2011- June 2021

5.     Hiring Committee (Spousal Assistant Professor Position, Dept. of Mathematics): December 2018

6.     Past Director of the Winnipeg Institute for Theoretical Physics: January 2017- December 2018

7.     Faculty of Science Executive Council, July 2016- June 2018

8.     Hiring Committee (Spousal Assistant Professor Position, Dept. of Mathematics): March 2017

9.     Hiring Committee (Assistant Professor Position, Dept. of Statistics): August 2015- March 2016

10.  Director of the Winnipeg Institute for Theoretical Physics: January 2015- December 2016

11.  Hiring Committee (Assistant Professor Position, Dept. of Mathematics): October 2014- April 2015

12.  Director Elect of the Winnipeg Institute for Theoretical Physics: January 2013- December 2014

 

As the Director of the Winnipeg Institute for Theoretical Physics (WITP) in 2015 and 2016, I was in charge of all administrative aspects of the institute, e.g.

·       organizing all WITP seminars at the University of Manitoba (8 to 10 per year);

·       approving all WITP expenses and balancing the budget; 

·       organizing a WITP summer research symposium at the end of August at which undergraduate and graduate students from the U of  Manitoba and U of Winnipeg  presented their work to other students and professors;

·       preparing the WITP annual report in December, which included all the research activities of the WITP and its members (publications, talks, etc.); 

·       maintaining the website of the WITP.

 

As the Past Director of WITP, I was one of three judges selecting the best PhD thesis in theoretical Physics for the DTP-WITP PhD Thesis Award; in 2017 there were 6 excellent submissions, totaling more than 1000 pages. Moreover, I was a co-organizer of a two-day WITP summer research symposium on July 31 and August 1 (2017) at the University of Manitoba to which we invited three well-established theoretical physicists and a mathematician (my collaborator from the University of Concepcion in Chile) as guest speakers to give one or two lectures each at the symposium.

 

As the Director Elect of WITP (2013 and 2014), I worked with the Director (A. Frey- University of Winnipeg) on preparing the 5-year review of WITP and on securing a significant increase in funding for the institute from the three universities in Manitoba (University of Manitoba, University of Winnipeg and Brandon University).

 

e.     Conference Organization and International Committees

      1. Member of the Scientific Committee for Numerical Computations: Theory and Algorithms (NUMTA2023) which will be held in Calabria, Italy (June 14-20, 2023)

2.     Member of the Scientific Committee for the 15th International Conference on p-Adic Analysis and Dynamical Systems to be held in Poznan, Poland (July 3-8, 2023)

3.     Member of the International Organizing Committee, Eighth International Conference on p-adic Mathematical Physics and its Applications, Online Conference (May 17-28, 2021)

4.     Member of the Scientific Committee for the 15th International Conference on p-Adic Functional Analysis (Poznan, Poland, July 6-10, 2020)- This was postponed to a later date due to COVID-19

5.     Member of the International Advisory Committee for the Seventh International Conference on p-Adic Mathematical Physics and its Applications (Covilha, Portugal, September 30- October 4, 2019)

6.     Member of the Scientific Committee for NUMTA2019: Numerical Computations: Theory and Algorithms (Crotone, Italy, June 15-21, 2019)

7.     Member of the Scientific Committee for the 15th International Conference on p-Adic Functional Analysis (Poznan, Poland, July 2-6, 2018)- This was postponed to July 6-10, 2020.

8.     Member of the Scientific Committee for the 14th International Conference on p-Adic Functional Analysis (Aurillac, France, July 4-9, 2016)

9.     Member of the Scientific Committee for NUMTA2016: Numerical Computations: Theory and Algorithms (Pizzo Calabro, Italy, June 20-24, 2016)

10.  Member of the Scientific Committee for the 13th International Conference on p-Adic Functional Analysis (University of Paderborn, Germany, August 12-16, 2014).

11.  Organizer of a special session on non-Archimedean Functional Analysis at the Seventh Conference on Function Spaces, Southern Illinois University at Edwardsville, May 20-24, 2014.

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

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

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

15.  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)

f.      Refereeing and Review services

1.     External reviewer of a PhD thesis in Mathematics, Universidad de la Frontera, Chile (2021)

2.     External reviewer for candidates for a tenure-track assistant professor position, Department of Mathematics, Linnaeus University, Sweden (2020)

3.     Reviewer for Zantrablatt Math and Mathematical Reviews (more than 25 papers since 2009)

4.     Referee of several papers for Math journals (more than 20 papers since 2008.)

5.     Referee of 6 research proposals (2 for Qatar Foundation, 2 for the Austrian Science Fund, 1 for BIRS, and 1 for CONICYT- Chile.)

 

g.     Community Service: Member of the Board of Directors for Campus Daycare May 2011- August 2014.

 

h.     Outreach: Volunteered at info days every year (2009-2014).

 

 

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

 

a.     Graduate Students Advisory Committees

1.     2003-2004: Javid Siddique; M.Sc. in Mathematics.

2.     2003-2004: Duygu Inceoz; M.Sc. in Mathematics.

 

b.     Thesis External Examiner

1.     2008: Dodzi Attimu; PhD in Mathematics; Howard University (US).

 

c.     Departmental Committees

1.     2007-2008

        1. Graduate Committee (Chair)
        2. Personnel Committee
        3. Curriculum Committee

2.     2006-2007

        1. Graduate Committee (Chair)
        2. Colloquium Committee
        3. Computer Affairs Committee
        4. Curriculum Committee

3.     2005-2006

        1. Graduate Committee (Chair)
        2. Colloquium Committee
        3. Curriculum Committee

4.     2004-2005

        1. Graduate Committee (Chair)
        2. Colloquium Committee
        3. Curriculum Committee
        4. Upper Division Curriculum Committee

5.     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

1.     2007-2008

        1. University Research Council
        2. Evidence of English Proficiency and Minimum TOEFL Requirements Committee
        3. College of Arts & Sciences Graduate Studies Committee

2.     2006-2007

        1. University Research Council
        2. Evidence of English Proficiency and Minimum TOEFL Requirements Committee
        3. College of Arts & Sciences Graduate Studies Committee

3.     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

o      The University of Manitoba and the University of Manitoba Faculty Association’s 2018 Merit Award in Teaching, Service and Research, May 29, 2019.

o      Ian C. P. Smith Integrated Science Faculty Scholar at the University of Manitoba for two consecutive terms: July 1, 2017- June 30, 2020 and July 1, 2020- June 30, 2022.

o      Nominated for the J. S. Frame Teaching Excellence Award for Faculty, Spring 2003, Department of Mathematics, Michigan State University.

o      Nominated for the J. S. Frame Teaching Excellence Award for Faculty, Spring 2002, Department of Mathematics, Michigan State University.

o      Hariri Foundation Fellowship to study in the USA (M.Sc.: 1988-1990 and PhD: 1993-1997); see this article, p.4.

o      Member of The Honor Society of Phi Kappa Phi.

o      Member of Phi Beta Delta, the Honor Society for International Scholars.

o      BS in Physics with High Distinction, American University of Beirut, June 1988.

o      Philip Hitti Award, American University of Beirut, June 1988, for graduating with the highest average in the Faculty of Arts and Sciences.

o      Dean's Honor List, faculty of Arts and Sciences, American University of Beirut, 1986-1988.

o      Malcolm Kerr Award, American University of Beirut, 1986-1988.

Professional Societies

o      American Mathematical Society

o      Canadian Association of Physicists

o      Canadian Mathematical Society

Languages

o      Arabic: fluent (reading, speaking and writing).

o      English: fluent (reading, speaking and writing).

o      French: good (reading, speaking and writing).