# Department of Mathematics and Statistics

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- ItemLp spaces of operator-valued functions(2021) Ramsey, Christopher; Reeves, AdamWe define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued Lp space that is shown to be complete. Various other norm candidates are considered as well as generalizations of H¨older’s inequality to this new context.
- ItemBistochastic operators and quantum random variables(2022) Plosker, Sarah; Ramsey, ChristopherGiven a positive operator-valued measure ν acting on the Borel sets of a locally compact Hausdorff space X, with outcomes in the algebra B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H, one can consider ν-integrable functions X → B(H) that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.
- ItemEffect of velocity and diffusion functionality on nonlinear mass transfer mechanisms in solvent oil recovery(2019) Lorimer, Shelley; Artymko, TimothyLiterature has indicated that, experimentally, solvent fronts in hybrid solvent recovery processes progress more rapidly than what can be predicted using current approximations and more rapidly than thermal processes alone. Research using finite differences to model the nonlinear advection, diffusion and dispersion (ADD) equation suggests that nonlinear mass transfer effects are important in predicting the rate of solvent advance. Nonlinearities can be ascribed to both diffusion and flow velocity functionality. Earlier work using linear concentration dependent diffusion and log-linear velocity behaviour confirmed the importance of nonlinear effects when compared to linear theory that uses constant diffusion, dispersion and velocity coefficients. The mathematical nature of the nonlinear ADD equation further suggests that the shape of concentration dependent diffusion and flow velocity will affect the shape of the solvent concentration profiles, and influence the rate of propagation of the solvent front. This research focuses on results obtained using finite differences to explore the effects of various diffusion and velocity functionalities that affect the solvent rate propagation using a nonlinear ADD equation. The results obtained from this analysis indicate that these functionalities determine the shape of the solvent concentration profile. The concentration dependent diffusion and velocity functions were chosen according to recent literature which proposes experimentally obtained functions to more accurately model solvent penetration in the media. Preliminary results from this study suggest that the velocity functionality has more influence on the process at both the lab and field scales for the parameters considered in this study. The shapes of the concentration profiles are affected by both diffusion functionality and velocity functionality.
- ItemPeriodic and solitary wave solutions for the one-dimensional cubic nonlinear Schrodinger model(2022) Bica, Ion; Mucalica, AnaUsing a similar approach as Korteweg and de Vries, [19], we obtain periodic solutions expressed in terms of the Jacobi elliptic function cn, [3], for the self-focusing and defocusing one-dimensional cubic nonlinear Schrodinger equations. We will show that solitary wave solutions are recovered through a limiting process after the elliptic modulus of the Jacobi elliptic function cn that describes the periodic solutions for the self-focusing nonlinear Schrodinger model.
- ItemHypercylindrical art(2022) Bica, IonEvery form and shape that surrounds us has a purpose, and mathematics helps in understanding it. Mathematics provides a source of abstract forms, i.e., mathematical structures, which are deliberately emptied of any content and therefore adaptable to any content. Art History is a testimony of great artists who adapted abstract mathematical forms, giving them artistic content. The article was inspired by a piece of art that I purchased of a talented emerging Ecuadorian artist, Tanya Zevallos. Her “Cylindrical Abstract Construction #124” artwork inspired me to look at the mathematical perspective that she gave in her art and construct further abstract aesthetics using cylinders.