The Galilee Research Center for Applied Mathematics
ORT Braude College
Complex Dynamical Systems
Control Theory, Differential Games and Optimization
Geometric Function Theory
Operator Theory and Nonlinear Analysis
Partial Differential Equations and General Relativity
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Department of Mathematics
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Approximate Convexity and an Edge-Isoperimetric Estimate
Prof. Vsevolod F. Lev, University of Haifa - Oranim, Israel
Mathematical model for cell motility driven by active gel
Dr. Ronen Avni, Technion / ORT Braude College
Cell crawling is a highly complex integrated process involving three distinct activities: protrusion adhesion and contraction, and also three players: the plasma membrane (car body), the actin network (engine) and the adhesion points (clutch). The actin network consists of actin polymers and many other types of molecules, e.g. molecular motors, which dynamically attach to and detach from the network, making it a biological gel. Furthermore, energy is consumed in the form of ATP due to both the activity of molecular motors and the polymerization at the filament tips; thus the system is far from thermodynamic equilibrium.
These characteristics make the above system unique and responsible for a wide range of phenomena (different force-velocity relationships) and behaviors (contraction, elongation, rotation, formation of dynamic structures).
Like the story on the blind men and the elephant, previous works considered only parts of the complex process, neglecting other sub-processes, or using unrealistic assumptions. Our goal was to derive a mathematical model for the whole system that can predict the rich variety of behaviors. For this purpose we had to identify the major players and integrate previous works into a one coherent mathematical model with (almost) no arbitrary constraints, adding our own mathematical description where needed. The model we
derived consists of several temporal and spatial scales, from molecular processes e.g. capping / branching to macro processes; furthermore, we used a hydrodynamic approach, hence could relate local dynamic events on
the boundary to the bulk inside the domain.
We focused on the processes near the leading edge that drive the system, i.e. the complexity comes in the b.c., and termed this filaments-membrane dynamics “the polymerization machinery”.
In my talk I will describe the mathematical model we derived and its relation to previous works. I will also describe a proprietary numerical simulation we derived for a free-surface flow of complex fluid in arbitrary geometries. Finally I will discuss open questions and opportunities in this line of research
November 26, 2013
The effects of host diversity on vector-borne disease: The conditions under which diversity will amplify or dilute the disease risk
Dr. Ezer Miller, The Gertner Institute, Israel
Multihost vector-borne infectious diseases form a significant fraction of the global infectious disease burden. In this study we explore the relationship between host diversity, vector behavior, and disease risk. To this end, we have developed a new dynamic model which includes two distinct host species and one vector species with variable preferences. With the aid of the model we were able to compute the basic reproductive rate,
, a well-established measure of disease risk that serves as a threshold parameter for disease outbreak. The model analysis reveals that the system has two different qualitative behaviors: (i) the well-known dilution effect, where the maximal
is obtained in a community which consists a single host (ii) a new amplification effect, denoted by us as diversity amplification, where the maximal
is attained in a community which consists both hosts.
The model analysis extends on previous results by underlining the mechanism of both, diversity amplification and the dilution, and specifies the
exact conditions for their
. We have found that diversity amplification occurs where the vector prefers the host with the highest transmission ability, and dilution is obtained when the vector does not show any preference, or it prefers to bite the host with the lower transmission ability. The mechanisms of dilution and diversity amplification are able to account for the different and contradictory patterns often observed in nature (i.e., in some cases disease risk is increased while in other is decreased when the diversity is increased).
Implication of the diversity amplification mechanism also challenges
current premises about the interaction between biodiversity, climate change, and disease risk and calls for retrospective thinking in planning intervention policies aimed at protecting the preferred host species.
November 12, 2013
Statistical Inference for Systems of Differential Equations
Dr. Ittai Datner, Department of Statistics, University of Haifa
Different phenomena in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their "true" value is thus required.
Motivated by a model for the biological process of blood coagulation, we focus on the class of nonlinear ordinary differential equations that have some linear structure in their parameters. If the system is nonlinear, an analytic solution to it usually does not exist. In such cases, parameter estimation methods such as MCMC or the non-linear least squares require the repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimizes the objective function.
In this talk we present a novel approach that bypasses the heavy computational task of numerical integration, avoids the estimation of slopes, and does not require searching the parameter space for a minimum. We develop two direct estimators for different experimental setups, discuss their statistical properties and illustrate their practical performance. In addition, a necessary and sufficient condition for identifiability of parameters will be presented.
Based on joint work with Chris A. J. Klaassen, Korteweg-de Vries Institute for Mathematics, University of Amsterdam.
October 29, 2013
A computational model of the human thyroid
Dr. Ludmila Ioffe, Department of Mathematics, ORT Braude College
The thyroid, the largest gland in the endocrine system, secretes hormones that regulate homeostatic functions within the body and promote normal growth and development. Recently, a detailed nonlinear computational model of the thyroid gland has been derived and used to explain clinical observations regarding the thyroid gland’s ability to maintain its hormonal secretion target in the face of uncertain dietary iodine intake levels. In this paper we reduce the model to an eight-order dynamical system, and analytically determine that a supercritical Hopf mechanism governs loss of stability of thyroid equilibrium, culminating with periodic self-excited oscillations beyond the critical threshold. In order to investigate model sensitivity to parameter uncertainty and orbital stability of periodic thyroid dynamic, we harmonically perturb both iodine intake and the level of thyrotropin, the thyroid-stimulating hormone. We numerically construct a comprehensive bifurcation structure that includes both periodic and subharmonic mode-locked solutions embedded within a set of quasiperiodic tori. An increase in the perturbation parameter magnitude reveals a similar and structurally stable bifurcation structure. Thus, the analysis of our nonlinear thyroid model shows that the gland can exhibit both equilibrium and periodic limit-cycle behavior which in-turn can lose its orbital stability due to harmonic perturbation.
July 11, 2013
Time-frequency integrals and the stationary phase method in problems of waves propagation from moving sources
Prof. Vladimir Rabinovich,
July 11, 2013
Local existence for the linearized compressible Euler equation with a free boundary
Prof. Uwe Brauer,
Departamento de Matem´atica Aplicada, Universidad Complutense, Madrid
June 10, 2013
The Geometric Study of the Hamiltonian Framework of Newton Equations
Prof. Gabriel Ben-Simon, ETH, Zurich
An equivalent formulation to Newton's Equations, of classical mechanics, is the Hamiltonian-Formulation, which is essentially a system of PDEs.These equations can not be solved in general thus, a qualitative study is requested. One of the ways to develop such a qualitative study (of the Hamiltonian PDEs) is by geometrizing the problem. Such a geometric approach appears as the study of the group of "Hamiltonian motions" which is a subfield of symplectic geometry. In my talk I will give a light introduction to this study, and I will present its main questions.
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