Department of Mathematics and Statistics

Permanent URI for this collection

Work by the faculty and students of the Department of Mathematics and Statistics


Recent Submissions

  • Item
    Technological Tools to Learn and Teach Mathematics and Statistics
    (Association for Computer Science and Telecommunications, 2012-02) Mesanovic, Mujo
    The blended learning method of teaching could be applied within the learning of mathematics and statistics. There are many different ways to obtain such learning, however one of the beneficial ways is to use MyLab and Mastering. MyLab and Mastering is blackboard website managed by the Pearson Publication Company which is used to enhance student learning in mathematics and statistics embedded within the current technological world. This paper describes the use of the MyLab and Mastering website from three points of view: instructor, students and administrative. This paper portrays the results of a recent study, where 120 students were enrolled in mathematics and statistics classes and used MyLab to improve their learning outcomes. Result of the case study are reported and analyzed. In addition suggestions are provided how to establish almost cost-free distance learning environments for mathematics and statistics.
  • Item
    The Boyle–Romberg trinomial tree, a highly efficient method for double barrier option pricing
    (MDPI, 2024-03-24) Leduc, Guillaume
    Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.
  • Publication
    Effects of mechano-electrical feedback on the onset of alternans: A computational study
    (AIP, 2019-06) Hazim, Azzam; Belhamadia, Youssef; Dubljevic, Stevan
    Cardiac alternans is a heart rhythm instability that is associated with cardiac arrhythmias and may lead to sudden cardiac death. The onset of this instability, which is linked to period-doubling bifurcation and may be a route to chaos, is of particular interest. Mechano-electric feedback depicts the effects of tissue deformation on cardiac excitation. The main effect of mechano-electric feedback is delivered via the so-called stretch-activated ion channels and is caused by stretch-activated currents. Mechano-electric feedback, which is believed to have proarrhythmic and antiarrhythmic effects on cardiac electrophysiology, affects the action potential duration in a manner dependent on cycle length, but the mechanisms by which this occurs remain to be elucidated. In this study, a biophysically detailed electromechanical model of cardiac tissue is employed to show how a stretch-activated current can affect the action potential duration at cellular and tissue levels, illustrating its effects on the onset of alternans. Also, using a two-dimensional iterated map that incorporates stretch-activated current effects, we apply linear stability analysis to study the stability of the bifurcation. We show that alternans bifurcation can be prevented depending on the strength of the stretch-activated current.
  • Publication
    A Simulation Study of the Role of Mechanical Stretch in Arrhythmogenesis during Cardiac Alternans
    (Biophysical Society, 2021-01) Hazim, Azzam; Belhamadia, Youssef; Dubljevic, Stevan
    The deformation of the heart tissue due to the contraction can modulate the excitation, a phenomenon referred to as mechanoelectrical feedback (MEF), via stretch-activated channels. The effects of MEF on the electrophysiology at high pacing rates are shown to be proarrhythmic in general. However, more studies need to be done to elucidate the underlying mechanism. In this work, we investigate the effects of MEF on cardiac alternans, which is an alternation in the width of the action potential that typically occurs when the heart is paced at high rates, using a biophysically detailed electromechanical model of cardiac tissue. We observe that the transition from spatially concordant alternans to spatially discordant alternans, which is more arrhythmogenic than concordant alternans, may occur in the presence of MEF and when its strength is sufficiently large. We show that this transition is due to the increase of the dispersion of conduction velocity. In addition, our results also show that the MEF effects, depending on the stretch-activated channels’ conductances and reversal potentials, can result in blocking action potential propagation.
  • Publication
    Efficiency of semi-implicit alternating direction implicit methods for solving cardiac monodomain model
    (Elsevier, 2021-01) Belhamadia, Youssef; Rammal, Zeinab
    It is well known that numerical simulations of the cardiac monodomain model require fine mesh resolution, which increases the computational resources required. In this paper, we construct three operator-splitting alternating direction implicit (ADI) schemes to efficiently solve the nonlinear cardiac monodomain model. The main objective of the proposed methods is to reduce the computational time and memory consumed for solving electrocardiology models, compared to standard numerical methods. The proposed methods have second-order accuracy in both space and time while evaluating the ionic model only once per time-step. Several examples using regular wave, spiral wave reentry, and nonsymmetrical scroll wave are conducted, and the efficiency of the proposed ADI methods is compared to the standard semi-implicit Crank–Nicolson/Adams–Bashforth method. Large-scale two- and three-dimensional simulations are performed.
  • Publication
    A projection scheme for phase change problems with convection
    (Elseiver, 2022-01) Haddad, M. El; Belhamadia, Youssef; Deteix, J.; Yakoubi, D.
    Numerical modeling of phase change problems with convection is known to be computationally expensive. The main challenge comes from the coupling between Navier–Stokes and heat energy equations. In this paper, we develop a new scheme for phase change problems based on a projection method. The proposed method reduces the size of the system by splitting the temperature, the velocity, and the pressure fields while preserving the accuracy of the simulations. A single-domain approach using a variant of the enthalpy-porosity formulation is employed. Incompressible Navier–Stokes problem with Boussinesq approximation for thermal effects in solid and liquid regions is considered. We regularize the discontinuous variables such as latent heat and material properties by a continuous and differentiable hyperbolic tangent function. The robustness and effectiveness of the proposed scheme are illustrated by comparing the numerical results with numerical and experimental benchmark.
  • Publication
    Efficiency of parallel anisotropic mesh adaptation for the solution of the bidomain model in cardiac tissue
    (Elsevier, 2022-04) Belhamadia, Youssef; Briffard, Thomas; Fortin, André
    Electrocardiology models are nonlinear reaction–diffusion type systems, where the numerical simulation requires extremely fine meshes to accurately compute the heart’s electrical activity. Anisotropic mesh adaptation methods have been proven to be efficient for simulating cardiac dynamic by many authors and showed a considerable improvement in the numerical accuracy while reducing the computational expenses. However, the efficiency of these techniques in parallel computing environments has not been shown yet, especially when compared to the performance of parallel uniform meshes. In this paper, we demonstrate the efficiency of a parallel anisotropic mesh adaptation method for the solution of the bidomain model in cardiac tissue. The technique is based on an efficient error estimator appropriate for second or higher order numerical solutions. To demonstrate the effectiveness of the developed methodology, comparisons between the numerical simulations on parallel adapted meshes with those on parallel uniform meshes are presented. The computational efficiency is assessed by computing spiral and scroll waves in cardiac tissue.
  • Publication
    Numerical modelling of hyperbolic phase change problems: Application to continuous casting
    (Elsevier, 2023-03) Belhamadia, Youssef; Cassol, Guilherme Ozorio; Dubljevic, Stevan
    Heat diffusion processes are generally modeled based on Fourier’s law to estimate how the temperature propagates inside a body. This type of modeling leads to a parabolic partial differential equation, which predicts an infinite thermal wave speed of propagation. However, experimental evidence shows that diffusive processes occur with a finite velocity of thermal propagation in many applications. In this paper, we develop a mathematical formulation to predict the finite speed of heat propagation in multidimensional phase change problems. The model generalizes the enthalpy formulation by adding a hyperbolic term. The governing equations are simulated by the finite element method. The proposed model is first verified by comparing numerical and experimental results illustrating the difference between the infinite and finite propagation velocity for heat inside biological tissues. Then, the results of the two and three-dimensional numerical solution of the continuous steel casting process are presented. We will illustrate that the effects of the initial conditions vanish faster when using the parabolic equation, while they persist in the hyperbolic modeling approach. The results demonstrate significant differences in the initial thermal dynamics and at the solid-liquid interface position when adding the hyperbolic term. The changes are more noticeable in the regions of the steel beam where rapid heat loss and, consequently, faster phase change occur.
  • Publication
    A mixed finite element method for nonlinear radiation–conduction equations in optically thick anisotropic media
    (Elsevier, 2023-09) Belhamadia, Youssef; Seaid, Mohammed
    We propose a new mixed finite element formulation for solving radiation–conduction heat transfer in optically thick anisotropic media. At this optical regime, the integro-differential equations for radiative transfer can be replaced by the simplified PN approximations using an asymptotic analysis. The conductivity is assumed to be nonlinear depending on the temperature along with anisotropic absorption and scattering depending on both the direction and location variables. The simplified PN approximations are enhanced by considering a diffusion tensor capable of describing anisotropic radiative heat transfer. In the present study, we investigate the performance of the unified and mixed formulations combining cubic P3, quadratic P2, and linear P1 finite elements to approximate the temperature in the simplified P3 model. To demonstrate the performance of the proposed methodology, three-dimensional examples of nonlinear radiation–conduction equations in optically thick anisotropic media are presented. The obtained numerical results demonstrate the accuracy and efficiency of the proposed mixed finite element formulation over the conventional unified finite element formulation to accurately solve the simplified P3 equations in anisotropic media.
  • Publication
    Liposomal Encapsulation of Chemotherapeutics Agents Combined with the Use of Ultrasound in Cancer Treatment
    (American Scientific Publishers, 2023-07) Almajed, Zeyad Mohamed; AlSawaftah, Nour Majdi; Sulieman, Hana; Husseini, Ghaleb
    Ultrasound (US) has numerous uses in the medical field, including imaging, tumor ablation, and lithotripsy; another interesting application of US in cancer therapy is as an external trigger in targeted drug delivery. Cancer-targeted drug delivery involves delivering chemotherapeutic drugs to tumor sites with a high degree of precision, which would minimize the adverse side effects experienced by patients. Several nanocarriers have been studied as possible nanocarriers; however, liposomes stood out from the rest because of their non-immunogenicity, amphiphilic nature, ease of functionalization, and stimuli-responsiveness. This review addresses the role of US in the synthesis of liposomes, its ability to induce localized and controlled drug release from liposomes, as well as the integration of US-induced release and US-imaging using liposomes as contrast agents utilizing thermal and/or mechanical effects.
  • Publication
    Modeling of the In Vitro Release Kinetics of Sonosensitive Targeted Liposomes
    (MDPI, 2022-12-05) Almajed, Zeyad Mohamed; Salkho, Najla; Sulieman, Hana; Husseini, Ghaleb
    Targeted liposomes triggered by ultrasound are a promising drug delivery system as they potentially improve the clinical outcomes of chemotherapy while reducing associated side effects. In this work, a comprehensive model fitting was performed for a large dataset of liposomal release profiles with seven targeting moieties (albumin, cRGD, estrone, hyaluronic acid, Herceptin, lactobionic acid, and transferrin) in addition to the control liposomes under ultrasound release protocols. Two levels of ultrasound frequencies were tested: low frequency (20 kHz) at 6.2, 9, and 10 mW/cm2 as well as high frequencies (1.07 MHz and 3 MHz) at 10.5 and 173 W/cm2. At a low frequency, Hixson–Crowell, Korsmeyer–Peppas, Gompertz, Weibull, and Lu–Hagen showed good fits to our release profiles at all three power densities. At high frequencies, the former three models reflected the best fit. These models will aid in predicting drug release profiles for future in vitro studies.
  • Publication
    Graph of Linear Transformations Over R
    (Springer, 2022-12-01) Badawi, Ayman; El-Ashi, Yasmine Ahmed
    In this paper, we study a connection between graph theory and linear transformations of finite dimensional vector spaces over R (the set of all real numbers). Let Rm, Rn be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rm into Rn. An equivalence relation ∼ is defined on L such that two elements f, k ∈ L are equivalent, f ∼ k, if and only if ker (f ) = ker (k). Let m, n ≥ 1 be positive integers and Vm,n be the set of all equivalence classes of ∼. We define a new graph, Gm,n, to be the undirected graph with vertex set equals to Vm,n, such that two vertices, [x] , [y] ∈ Vm,n are adjacent if and only if ker (x) ∩ ker (y) 6 = 0. The relationship between the connectivity of the graph Gm,n and the values of m and n has been investigated. We determine the values of m and n so that Gm,n is a complete graph. Also, we determine the diameter and the girth of Gm,n.
  • Publication
    Maximum principles and overdetermined problems for Hessian equations
    (ISTE Group, 2021) Enache, Cristian; Marras, Monica; Porru, Giovanni
    In this article we investigate some Hessian type equations. Our main aim is to derive new maximum principles for some suitable P-functions, in the sense of L.E. Payne, that is for some appropriate functional combinations of u(x) and its derivatives, where u(x) is a solution of the given Hessian type equations. To find the most suitable P-functions, we first investigate the special case of a ball, where the solution of our Hessian equations is radial, since this case gives good hints on the best functional to be considered later, for general domains. Next, we construct some elliptic inequalities for the well-chosen P-functions and make use of the classical maximum principles to get our new maximum principles. Finally, we consider some overdetermined problems and show that they have solutions when the underlying domain has a certain shape (spherical or ellipsoidal).
  • Publication
    Fractional Integrodifferentiation and Toeplitz Operators with Vertical Symbols
    (Birkhäuser, 2020) Karapetyants, Alexey; Louhichi, Issam
    We consider the so-called vertical Toeplitz operators on the weighted Bergman space over the half plane. The terminology “vertical” is motivated by the fact that if a is a symbol of such Toeplitz operator, then a(z) depends only on y = ℑz, where z = x + iy. The main question raised in this paper can be formulated as follows: given two bounded vertical Toeplitz operators Tλa and Tλb, under which conditions is there a symbol h such that TλaTλb=Tλh? It turns out that this problem has a very nice connection with fractional calculus! We shall formulate our main results using the well-known theory of Riemann–Liouville fractional integrodifferentiation.
  • Publication
    On the powers of quasihomogeneous Toeplitz operators
    (Springer, 2021) Aissa, Bouhali; Zohra, Bendaoud; Louhichi, Issam
    In this article, a fixed point iterative scheme involving Green's function is applied to reach a solution for the buckling of nano-actuators under nonlinear forces. Our solution is convergent. The nano-actuators problem under consideration is governed by a general type equation that contains nonlinear forces and integro-differential terms. The equation, we adopted and which governs the nano-actuators, is a nonlinear integro-differential BVP of fourth order. Our scheme enjoys important features such as high accuracy, robustness, and fast convergence. Numerical tests are performed and compared with other results that exist in the current literature.
  • Publication
    A Numerical Investigation of the Buckling of Doubly Clamped Nano-Actuators Governed by an Integro-Differential Equation
    (Springer, 2022) Yousef, Abdelrahman; Khoury, Suheil; Louhichi, Issam
    In this article, a fixed point iterative scheme involving Green's function is applied to reach a solution for the buckling of nano-actuators under nonlinear forces. Our solution is convergent. The nano-actuators problem under consideration is governed by a general type equation that contains nonlinear forces and integro-differential terms. The equation, we adopted and which governs the nano-actuators, is a nonlinear integro-differential BVP of fourth order. Our scheme enjoys important features such as high accuracy, robustness, and fast convergence. Numerical tests are performed and compared with other results that exist in the current literature.
  • Publication
    On 1-absorbing primary ideals of commutative rings
    (World Scientific, 2019-04-30) Badawi, Ayman; Celikel, Ece Yetkin
    Let R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements a,b,c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if a ring R is not a quasilocal, then a proper ideal I of R is a 1-absorbing primary ideal of R if and only if I is a primary ideal. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form Pn for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I1I2I3 ⊆ I for some proper ideals I1, I2, I3 of R, then I1I2 ⊆ I or I3 ⊆ √I.
  • Publication
    On Weakly 1-Absorbing Primary Ideals of Commutative Rings
    (World Scientific, 2022) Badawi, Ayman; Celikel, Ece Yetkin
    Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing primary ideal. A proper ideal I of R is called a weakly 1-absorbing primary ideal if whenever nonunit elements a, b, c ∈ R and 0 ≠ abc ∈ I, then ab ∈ I or c ∈ √I. A number of results concerning weakly 1-absorbing primary ideals and examples of weakly 1-absorbing primary ideals are given. Furthermore, we give the correct version of a result on 1-absorbing primary ideals of commutative rings.
  • Publication
    On n-semiprimary ideals and n-pseudo valuation domains
    (Taylor and Francis, 2020-08-14) Anderson, David F.; Badawi, Ayman