## Mathematical and Computational Science

Faculty and students at the Department of Physics are concerned with research in the area of theoretical molecular spectroscopy. In details, the theoretical calculation of the electronic states of a diatomic molecule via the ASSCF/MRCI (single and double excitations with Davidson correction) method using a correlation-consistent basis set. This calculation leads to the investigation of the potential energy curves, spectroscopic constants, the static and transition dipole moment, the Einstein coefficients, the spontaneous radiative life time and the emission oscillator strength. The ro-vibrational parameters, as well as the abscissas of the turning points and can be obtained for different electronic states with the canonical functions approach. These data are beneficial for investigating the feasibility of laser cooling and their applications along with other use for the scientific community.

Faculty and students in the program of mathematics are working on research related to both pure mathematics and applied mathematics. Theoretical and numerical tools based on computations made the program acquire a variety of research schemes and publications that do not only have impact on the program but also on other fields of science as well.

Group Representations: A professor in Algebra, Mohammad N. Abdulrahim, is currently working on representations of the braid group, whose impact on Quantum Computing is significant. Braids are correlated with the study of qudits and the exchange of the anyons inside them. Representations of the braid groups have been proposed as a fractional statistics of Anyons, and used in the topological models for quantum computing. Some MS and Ph.D. students are currently working on identifying the images of such braid group representations and studying their unitarity which play an important role in quantum physics. A recent paper, namely “The Hecke Algebra Representation of the complex reflection group G_7 is unitary”, has been recently accepted in the journal JP journal of Algebra, Number Theory and Applications. A 2- dimensional representation of the Hecke algebra was studied and a necessary and sufficient condition for an element in the kernel of the representation was determined. This is an important step in tackling the “faithfulness” question regarding unitary representations of the braid group.

Fluid Dynamics: An Associate professor, Toufic Al-Arwadi, and an Associate professor Emad Ashmawy, are currently working on theoretical and numerical analysis for some models in fluid dynamics. Analysis of several models (e.g. Navier- Stokes and Brinkman models) are studied theoretically and numerically. Dr. Emad Ashmawy has recently proposed some boundary value problems coming from the models and studied them analytically. Dr. Toufic Al-Arwadi has proposed two numerical schemes (implicit and explicit) to solve them numerically. Dr Emad is also interested in working on the theory of couple stress fluids and a recent puplication “Unsteady Stokes Flow of a Couple Stress Fluid around a Rotating Sphere with Slip” has appeared in The European Physical Journal Plus, 131: 175, 2016.

Inverse Problem and Application in Medical Imaging: Dr. Toufic, in collaboration with Ali Zaart, professor in Computer Science, also works on using existent methods in partial differential equations to detect breast cancer. A recent publication “Analysis of D-bar methods from complex conductivity” has been accepted in Journal of Analysis and Applications, 2016.

Applied Statistics: An Assistant Professor, Noura Yassine, studies probabilistic methods for several cases in applied sciences. A recent publication “The effects of approximating a probabilistic process using the renewal reward theorem and the optimal solution of an EQQ model with quality” has been published in International journal of Mathematics and Computers in Simulation, V. 10, pp. 354-360, 2016.

A future plan includes integrating the theory of braids in the theory of quantum computing benefiting from the linear unitary representations of various dimensions already determined. It also includes coming up with different models in fluid dynamics which have influences on blood flow and other fluids. Also, the orientation of applied statistics will get more involved into various aspects in life, where statistical models are constructed.