Mathematical Sciences

Friday, October 16

3:00-4:00pm

 Zoom Meeting ID: 947 0114 5457

Stanley Snelson

Department of Mathematical Sciences

 Florida Institute of Technology

Abstract: Kinetic theory seeks to understand the physical properties of matter by studying statistical averages of its constituent particles. Mathematically, this leads to a differential equation for the particle density function (i.e. the density of particles at time t, location x, and velocity v). This talk will focus on two classical kinetic differential equations: the Boltzmann equation (1872) which models diffuse gases, and the Landau equation (1936) which models plasmas. Despite their long history, these equations have proven difficult to understand mathematically, and the question of global existence vs. breakdown remains open (in the general case) for both equations. I will give an overview of recent progress on these equations, focusing especially on the program of conditional regularity, which gives physically meaningful conditions under which the solution can be extended past a given time. The talk will end with a discussion of some promising directions for future research. 

Friday, November 6

3:00-4:00pm

 Zoom Meeting ID: 947 0114 5457

Jian Du

Department of Mathematical Sciences

Florida Institute of Technology

Abstract: The stability of a platelet thrombus under flow depends strongly on the local hemodynamics and on the thrombus’ structural properties such as porosity, permeability, and elasticity. We develop a two-phase continuum model to investigate the biomechanics of thrombus stability in fluid channels. It is among the few existing models that are capable of explicitly tracking the formation and breaking of interplatelet molecular bonds, which directly determine the viscoelastic property of the thrombus and govern its ability to resist fluid drag. We characterize the stability/fragility of thrombi for various flow speeds, porosities, bond concentrations, and bond types.

Friday, November 20

3:00-4:00pm

Zoom Meeting ID: 947 0114 5457

Antonnette Gibbs 

Department of Mathematical Sciences

Florida Institute of Technology

Abstract: Socio-critical modeling is one of the six modeling perspectives in mathematics education. The socio-critial perspective on mathematical modeling is distinguished by its focus on:

• Critical view of mathematical models and the consequences of these models in society;
• Applying modling to real-life problems (social, political, economic, etc.);
• Empowerment of students and development of reflexive discourse among students.

The soci-critical modeling is constituted in practice through student reglexive discussions during mathematical modeling activities; however, little guidance can be found in the literature on how to design and plan slassroom activities that would stimulate such reflexive discussions.

This talk will report findings from a project that examined student reflexive discussions about the role of mathematics in society based on a designed pedagogical tool. The project involves 27 community students' collaborative mathematics modeling activites, and student interation data was analyzed by the Constuctivist Grounded Theory (CGT) approach. The talk will highlight four concepts and an emerging theory from data along with key examples from students discourses. I will close the talk by contextualizing this project in a broader body of work focused on mathematics classroom projects that foster intellectually productive and critial citizens. This talk is based on findings from my dissertation research (Ph.D. advisor: Dr. Joo Young Park, Florida Institute of Technology).

Friday, September 14

3:00 - 4:00pm

The MAC 

Dr. Ugur G. Abdulla

Department of Mathematical Sciences

Florida Institute of Technology 

Abstract: In this talk I am going to discuss the inverse Electrical Impedance Tomography (EIT) problem or Calderon problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. The inverse EIT problem presents an effective mathematical model of breast cancer detection based on the experimental fact that the electrical conductivity of malignant tumors of the breast is significantly different from conductivity of the normal tissue. I am going to introduce a mathematical model of the inverse EIT problem as a PDE constrained optimal control problem in a Sobolev-Besov spaces framework, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. Some recent results on the existence of the optimal control, Frechet differentiability in the Besov space setting, derivation of the he formula for the Frechet gradient, optimality condition, and extensive numerical analysis in the 2D case through implementation of the gradient method in Banach spaces will be presented. Talk will end with the formulation of some major open problems and the perspectives of future advance. 

Friday, October 5

3:00 - 4:00pm

The MAC

Dr. Yalchin Efendiev

Department of Mathematics

Texas A&M University 

Abstract: In this talk, I will discuss several data integration techniques in multiscale simulations. I will give a brief overview of multiscale simulation concepts that will be used. These multiscale techniques are designed for problems when the coarse grid does not resolve scales and contrast. I will describe the relation between multiscale and upscaling methods. I will describe three data integration techniques. The first one, Bayesian multiscale modeling, will sample basis functions and incorporate available data. In the second approach, we will use deep learning techniques to design and modify existing multiscale methods in the presence of data and nonlinearities.

 Friday, October 12

3:00 - 4:00pm

The MAC

 Dr. Alexandr Tamasan

Department of Mathematics

University of Central Florida

Abstract: In this talk I will present the inverse hybrid problem of CDII, where the electrical conductivity of body is to be recovered from of the magnitude of one current density field. Physically we use a connection between the current density field generated from the boundary, with interior measurements of the magnetization in a MRI machine. The basic idea is to first recover the voltage potential, which solves a generalized 1-Laplacian. Mathematical methods involved in solving this problem combines ideas from Riemannian geometry with geometric measure theory, and touch on some algebraic topology. 

 Friday, February 22

3:00 - 4:00pm

The MAC

 Dr. Benjamin F. Akers

Department of Mathematics & Statistics

Air Force Institute of Technology

Title:  Asymptotics and Numerics for Modulational Instabilities of Traveling Waves.

Abstract:  The spectral stability problem for periodic traveling waves for water wave models is considered. The structure of the spectrum is discussed from the perspective of resonant interaction theory. Modulational asymptotic expansions are used to predict the location of instabilities in frequency-amplitude space. These predictions explain numerical results in [1]. Asymptotics results are presented in the potential ow equations [2] as well as weakly nonlinear models [3]. The asymptotic predictions are compared to the results of a direct numerical simulation of the modulational spectrum.

References

 [1]  Nicholls, David P., Spectral data for travelling water waves: singularities and stability, Journal of Fluid Mechanics, 624 (2009), 339-360.

[2] Akers, Benjamin F., Modulational Instabilities of periodic traveling waves in deep water, Physica D: Nonlinear Phenomena, 300 (2015), 26-33.

[3] Akers, Benjamin F. and Milewski, Paul A., A Model Equation for Wavepacket Solitary Waves Arising from Capillary-Gravity Flows, Studies in Applied Mathematics, 122 (2009), 249-274.

Friday,  April 12

3:00 - 4:00pm

The MAC

Adam Prinkey

Department of Mathematical Sciences

Florida Institute of Technology

Title:   Qualitative Analysis of the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption

Abstract:  This talk presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption u_t=(|(u^m)_x|^p-1(u^m)_x)_x-bu^β. The nonlinear partial differential equation above is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in with slow (mp>1) or fast (0<mp<1) regime and nonlinear state dependent reaction (b>0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters m, p, b, and β, and asymptotics of the initial function near its support. In the fast diffusion regime strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The proof of the resulrs is based on rescaling laws for the nonlinear PDEs and blow-up techniques for the identification of the asymptotics of the solution near the interfaces, construction of barriers using special comparison theorems in irregular domains with characteristic boundary curves.

 Friday, April 19

3:00 - 4:00pm

The MAC

 Dr. Robert Talbert

Department of Mathematics

Grand Valley State University

 Title: Flipped learning in theory and practice for mathematics

Abstract: Flipped learning is a pedagogical model in which learners get first contact with new ideas through guided and structured self-learning activities prior to group meetings, allowing significantly more attention to be paid during class meetings to more advanced topics through challenging active learning tasks done collaboratively. Flipped learning combines the best traditions of tutorial and case study methods with modern technologies and evidence-based teaching practices, to help create highly effective learning environments for all learners. In this talk, we will trace the origins of flipped learning, present a framework for flipped learning based in self-determination theory, and illustrate real-life examples from applications to mathematics courses. 

 Friday, April 26

3:00 - 4:00pm

The MAC

Roby Poteau

Department of Mathematical Sciences

Florida Institute of Technology

Title: Identification of Parameters in Systems Biology

Abstract: Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components.  One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. This dissertation explores the inverse problem for the identification of the finite dimensional set of parameters for systems of nonlinear ordinary differential equations (ODEs) arising in systems biology. Two numerical methods are implemented. The first method combines the ideas of Pontryagin optimization or Bellman's quasilinearization with sensitivity analysis and Tikhonov's regularization. The method is applied to various biological models such as the classical Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate the quadratic convergence. The method proved to be extremely effective in moderate scale models of systems biology. The results are published in a recent paper in Mathematical Biosciences, 305(2018), 133-145. To address adaptation and scalability of the method for large-scale models of systems biology the modification of the method is pursued by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. The modified method is applied to benchmark model of three-step pathway modeled by 8 nonlinear ODEs with 36 unknown parameters and two control input parameters. The numerical results demonstrate the geometric convergence with minimum five data sets and with minimum measurements per data set. The method is extremely robust with respect to partial and noisy measurements, and in terms of required number of measurements for each components of the system. Optimal choice of the Tikhonov regularization parameter significantly improves convergence rate, precision and convergence range of the algorithm. Software package qlopt is developed for both methods and posted in GitHub. MATLAB package AMIGO2 is used to demonstrate advantage of qlopt over most popular methods/software such as lsqnonlin, fmincon and nl2sol.