The Eighth Congress of Romanian Mathematicians

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List of talks

I. Algebra and Number Theory

Special session: Local rings and homological algebra. Special session dedicated to Prof. Nicolae Radu

II. Algebraic, Complex and Differential Geometry and Topology

Special session: Geometry and Topology of Differentiable Manifolds and Algebraic Varieties

III. Real and Complex Analysis, Potential Theory

IV. Ordinary and Partial Differential Equations, Variational Methods, Optimal Control

Special session: Optimization and Games Theory

V. Functional Analysis, Operator Theory and Operator Algebras, Mathematical Physics

Special session: Spectral Theory and Applications in Mathematical Physics

Special session: Dynamical Systems and Ergodic Theory

VI. Probability, Stochastic Analysis, and Mathematical Statistics

VII. Mechanics, Numerical Analysis, Mathematical Models in Sciences

Special session: Mathematical Modeling of Some Medical and Biological Processes

Special session: Mathematical Models in Astronomy

VIII. Theoretical Computer Science, Operations Research and Mathematical Programming

Special session: Logic in Computer Science

IX. History and Philosophy of Mathematics

Mechanics, Numerical Analysis, Mathematical Models in Sciences 

(this list is in updating process)

University Politehnica of Bucharest, Romania
Title: Stability analysis of some equilibrium points in a complex model for blood cells’ evolution in CML (details)
The complex model considers the competition between the populations of healthy and leukemic stem-like short-term and mature leukocytes and the influence of the T-lymphocytes on the evolution of leukemia. Delay differential equations are used in a modified Mackey-Glass approach, with the consideration of asymmetric division and feedback functions for the action of the immune system. This research is focused on the linear stability analysis of equilibrium points. As the characteristic equations for some equilibrium points are very complex, the existence of a Lyapunov-Krasovskii functional is investigated. Also, treatment with Imatinib is introduced in the model and new stability properties are investigated. This research is supported by the CNCS Grant PNII-ID-PCE-3-0198 and by POSDRU/159/1.5/S/132395.
BALAN Vladimir
University Politehnica of Bucharest, Romania
Title: Spectral aspects of anisotropic metric models in the Garner oncologic framework (details)
%================================= % CMR-2015 * TALK * Balan-Stojanov %================================= \documentclass[11pt]{article}\begin{document}\thispagestyle{empty}\begin{abstract} % The present talk discusses three natural Finsler models, which naturally relate to the classical Garner dynamical system, which describes the evolution of the active and quiescent cancer cell populations. % The statistically fit metric structures are determined from the energy of the deformed field of the biological model, assuming that severe disease circumstances occur, and it is shown that the subsequently derived geometric objects are able to provide an evaluation of the overall cancer cell population growth.\par % The spectral characteristics of the Cartan tensor, the comparison between the $Z-$ and $H-$eigendata of the constructed Randers, $m-$th root and Euclidean structures, and the applicative advantages of the developed geometric models, are discussed.\\[2mm] % 2010 \textit{Mathematics Subject Classification:} 53B40, 53C60, 37C75, 65F30, 15A18, 15A69.\\[1mm] % \end{abstract}\end{document} %=================================
RIT (Rochester Institute of Technology), USA
Title: RIT's Cubesat Project (details)
Students and faculty members of the Space Exploration group (SPEX) at the Rochester Institute of Technology (RIT) have worked together on a CubeSat project focused on laser communications. Laser communications represent a major change in how spacecraft communications could be handled in the future and this is an important area of research in the space community. Our plan is to launch a satellite through the NASA CubeSat Launch Initiative and we have two main scientific goals: testing laser uplink technologies and developing better tracking algorithms. In this paper we will discuss the scientific merit of the proposal and the technical challenges regarding this mission.
University Duisburg-Essen and University A.I. Cuza of Iasi, Germany and Romania
Title: On the 6-parameter shell model derived from the three-dimensional Cosserat theory of elasticity (details)
In order to obtain a refined shell model, we start from a nonlinear Cosserat model for three-dimensional elastic bodies, with a specific form of the strain energy functional (see [1]). We perform the dimensional reduction by integration through the thickness and using a generalized plane stress condition, which allows to determine in closed form the expression for the thickness stretch and the nonsymmetric shift of the midsurface in bending. Thus, we derive a two-dimensional shell model which takes full account of initial curvature effects and involves 6 parameters (kinematical degrees of freedom) for each material point: 3 for translations and 3 for rotations. Within this new model we can express the strain measures with the help of shell strain and bending-curvature tensors which were previously introduced in the general nonlinear theory of 6-parameter shells (see e.g., [2,3]). In our model we obtain a specific form for the shell strain energy density and show that this functional satisfies the invariant properties required by the local symmetry group of isotropic solid shells, as stated in [2]. We also prove the existence of minimizers for 6-parameter elastic shells, in a general case including anisotropic behavior, under certain convexity assumptions on the energy functional [3]. An important special case of the general shell theory is the case of shells without drilling rotations, where the in-plane rotations (i.e. rotations about the shell filament) are neglected. For this type of shells one obtains a 5-parameter model (3 translational and 2 transverse-rotational degrees of freedom, i.e. a Reissner-type kinematics) which is widely used in engineering, since it covers most of the practical situations. In order to characterize this type of shells in the framework of the general shell theory, we prove a representation theorem for the strain energy functional of shells without drilling rotations [4]. Finally, we show that this new model can be successfully applied to describe and solve numerically some complex mechanical problems, such as the formation of wrinkles in a thin elastic sheet under shear [5]. \textbf{References} 1. P. Neff, M. B\^{\i}rsan, F. Osterbrink (2015): Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements, Journal of Elasticity, DOI 10.1007/s10659-015-9517-6. 2. V.A. Eremeyev, W. Pietraszkiewicz (2006): Local symmetry group in the general theory of elastic shells, Journal of Elasticity, vol. 85, 125-152. 3. M. B\^{\i}rsan, P. Neff (2014): Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations, Mathematics and Mechanics of Solids, vol. 19, 376-397. 4. M. B\^{\i}rsan, P. Neff (2014): Shells without drilling rotations: A representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory, International Journal of Engineering Science, vol. 80, 32-42. 5. O. Sander, P. Neff, M. B\^{\i}rsan (2015): Numerical treatment of a geometrically nonlinear planar Cosserat shell model, published first on arXiv: 1412.3668v2 , submitted.
"Simion Stoilow" Institute of Mathematics of the Romanian Academy, Romania
Title: A quasistatic frictional contact problem with normal compliance and unilateral constraint (details)
We consider a mathematical model describing the quasistatic process of frictional contact between an elastic body and a foundation. The contact is modeled by normal compliance and unilateral constraints, and the friction by a slip-dependent version of Coulomb's law. A weak formulation of the problem is derived and an existence result is proved by using arguments of incremental formulations, compactness and lower semicontinuity.
University of Bucharest & Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy, Romania
We linearize the equations of aerodynamics and express the jump of the pressure past the airfoil in terms of the solution of a hypersingular integral equation (the lifting surface equation). The great advantage of the lifting surface equation is that the domain of integration is the airfoil (more precisely its projection onto the Oxy - plane) and the influence of the vortex sheet is taken into account by means of the expresion of the kernel. After solving the lifting surface equation we calculate the aerodinamic coefficients (lift, drag and moment coefficients) and the pressure field.
University of Colorado Colorado Springs, USA
Title: Optimization and Control in Vascular Networks (details)
The cardiovascular system is designed to be a very robust system, meant to function under a wide range of external conditions. Understanding its dynamics in basal condition as well as in such dynamic states is desirable and several mathematical models have been developed to achieve this goal. In this talk we present a Boussinesq-type system for modeling the dynamics of pressure-flow in arterial networks, considered as a 1d spatial network. Numerical solutions of the system of PDE are obtained via discontinuous Galerkin schemes and are compared with simplified models based on particle-tracking arguments. We present several flow optimization studies, which include the geometry and size of the network as well as boundary control. Physiologically realistic control mechanisms are also tested in the context of these simplified models.
Università degli Studi di Milano, Italy
Title: Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. (details)
We consider a non-isothermal modified viscous Cahn-Hilliard equation. Such an equation is characterized by an inertial term and it is coupled with a hyperbolic heat equation from the Maxwell-Cattaneo's law. We analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with finite energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz-Simon inequality.
Universitatea de Stiinte Agricole si Medicina Veterinara "Ion Ionescu de la Brad" Iasi, Romania
Title: Rein's Model for the Restricted Eliptic Three-Body Problem with drag (details)
In this paper we present a generalization of Rein's model for the elliptic restricted three-body problem (ERTBP) by taking into account of a drag force. The equations of motion and the sationary points were established, and the linear stability of the equilibrium points were studied. Applications to the Earth-Moon system are considered, with the traiectories computed around the stable Lagrangian points.
COLLI Pierluigi
University of Pavia, Italy
Title: Non-smooth regularization of a forward-backward parabolic equation (details)
A variation of the Cahn-Hilliard equation is discussed. In the concerned system, a viscosity term, with a maximal monotone graph acting on the time derivative of the phase variable, replaces the usual diffusive contribution. The phase variable stands for the concentration of a chemical species and it evolves under the influence of a non-convex free energy density. For the chemical potential a non-homogeneous Dirichlet boundary condition is assumed. Existence and continuous dependence results are discussed. The talk reports on a joint work with E. Bonetti and G. Tomassetti.
CRACIUN Eduard - Marius
Universitatea OVIDIUS Constanta,, Romania
The considered problem is the antiplane deformation in prestressed and prepolarized piezoelectric crystals in equilibrium. The representation of the general solution is derived in terms of complex potentials for all piezoelectric crystal classes in which an antiplane state is possible. This generalizes earlier results obtained in respect of a specific crystal class. The general formulae are specialized to find the antiplane state generated by a Mode-III crack.
Université de Caen, France
Title: Fading regularization method for Cauchy problems associated with elliptic operators (details)
Data completion problems, also known as Cauchy problems, associated with elliptic equations are investigated. These problems consist of recovering the missing data on a part of the boundary of the solution domain from over-specified data on the remaining boundary. A regularization method is introduced based on an iterative algorithm whose regularization effect vanishes with respect to increasing the number of iterations. The principle, the continuous and discrete formulations of the method and numerical simulations are presented.
"Grigore T. Popa" University of Medicine and Pharmacy of Iasi, Romania
Title: Numerical simulations of a two noncompeting species chemotaxis model (details)
In this study we present the results of the numerical simulations of a two-species chemotaxis model. This model represents a regularized extension of the Patlak-Keller-Segel (PKS) system to the case of the chemotaxis motion of two noncompeting species that produce the same chemoattractant. We perform several experiments by applying a strong stability preserving (SSP) implicit-explicit Runge-Kutta method to study the behaviour of the obtained spiky solutions.
University of Maryland Baltimore County, USA
Title: Optimal Order Multigrid Preconditioners for Linear Systems Arising in the Semismooth Newton Method Solution Process of a Class of Control-Constrained Problems (details)
In this work we present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution process of certain control-constrained, quadratic distributed optimal control problems. Using a piecewise constant discretization of the control space, each semismooth Newton iteration essentially requires inverting a principal submatrix of the matrix entering the normal equations of the associated unconstrained optimal control problem, the rows (and columns) of the submatrix representing the constraints deemed inactive at the current iteration. Previously developed multigrid preconditioners by Draganescu [Optim. Methods Softw., 29 (2004), pp. 786-818] for the aforementioned sub matrices were based on constructing a sequence of conforming coarser spaces, and proved to be of suboptimal quality for the class of problems considered. Instead, the multigrid preconditioner introduced in this work uses non-conforming coarse spaces, and it is shown that, under reasonable geometric assumptions on the constraints that are deemed inactive, the preconditioner approximates the inverse of the desired submatrix to optimal order. The preconditioner is tested numerically on a classical elliptic-constrained optimal control problem and further on a constrained image-deblurring problem.
GALES Catalin Bogdan
Al. I. Cuza University of Iasi, Romania
Title: Dynamics of space debris: resonances and long term orbital effects (details)
Since the beginning of space exploration a large number of space debris accumulated in the vicinity of the Earth, from near atmosphere to the geosynchronous region. The impact of operative spacecraft or satellites with large enough space debris could result in a dramatic situation. Understanding the overall orbital evolution of space debris is essential for maintenance and control strategies, as well as for space debris mitigation. In this talk, we present a description of the dynamics of space debris in various resonances by using the Hamiltonian formalism. We consider a model including the geopotential contribution and we compute the secular and the resonant parts of the Hamiltonian function for each resonance. Determining the leading terms of the expansions in specific resonant regions, we explain the main dynamical features of each resonance in a very effective way. Then, by computing the Fast Lyapunov Indicators, we provide a cartographic study of each resonance, yielding the regular and chaotic behavior of the resonant regions. The study allows to determine easily the location of the equilibrium points, the amplitudes of the libration islands and the main dynamical stability features of each resonance. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the lunar and solar gravitational attractions and the solar radiation pressure.
Technical University of Iasi, Romania
Title: Mathematical insights and integrated strategies for the control of Aedes aegypti mosquito (details)
We propose and investigate a delayed model for the dynamics and control of a mosquito population which is subject to an integrated strategy that includes pesticide release, the use of mechanical controls and the use of the sterile insect technique (SIT). The existence of positive equilibria is characterized in terms of two threshold quantities, being observed that the ``richer" equilibrium (with more mosquitoes in the aquatic phase) has better chances to be stable, while a longer duration of the aquatic phase has the potential to destabilize both equilibria. It is also found that the stability of the trivial equilibrium appears to be mostly determined by the value of the maturation rate from the aquatic phase to the adult phase. A nonstandard finite difference (NSFD) scheme is devised to preserve the positivity of the approximating solutions and to keep consistency with the continuous model. The resulting discrete model is transformed into a delay-free model by using the method of augmented states, a necessary condition for the existence of optimal controls then determined. The particularities of different control regimes under varying environmental temperature are investigated by means of numerical simulations. It is observed that a combination of all three controls has the highest impact upon the size of the aquatic population. At higher environmental temperatures, the oviposition rate is seen to possess the most prominent influence upon the outcome of the control measures.
GHEORGHIU Calin - Ioan
Romanian Academy, T. Popoviciu Institute of Numerical Analysis, Cluj-Napoca, Romania
Title: From Separation of Variables to Multiparameter Eigenvalue Problems. Numerical Aspects. (details)
The main aim of this work is to show that the Jacobi-Davidson type (space) methods are fairly accurate and robust methods for solving algebraic multiparameter eigenvalue problems that are provided by separation of variables applied to various boundary value problems attached to some classical partial differential equations (Helmholtz, Laplace, Schroedinger, Mathieu, Lamé, etc.). The differential systems are discretized by spectral collocation method based on Chebyshev, Laguerre or Hermite functions. These methods also solve generalized algebraic eigenvalue problems with a singular second matrix. The first matrix in the pencil is nonsymmetric, full rank and ill-conditioned. Despite of these difficulties, and the large dimensions of the multiparameter problems, a specified region of the spectrum is computed accurately, as it is shown for matrices arising in some hydrodynamic stability problems. The methods also overcome the potentially severe problems associated with spurious unstable eigenvalues. A lot of numerical experiments are carried out and some pseudospectra and eigenfunctions are displayed for various aspect ratios of the geometrical domains.
University of Trento, Italy
We consider a model with age and space structure designed to describe the evolution of the supra-basal epidermis. The model, presented and analyzed in [1] [2], includes different types of cells (proliferating cells, differentiated cells, corneous cells, and apoptotic cells) that move with the same velocity so that the local volume fraction, occupied by the cells is constant in space and time. We investigate the well-posedness of the problem determining conditions for the existence of a moving boundary representing the surface of the skin. We also consider the stationary case of the problem, that takes the form of a quasi-linear evolution problem of first order. This case corresponds to the normal status of the skin. A numerical scheme to compute the solution of the model is proposed and analyzed. Simulations are provided for realistic values of the parameters.
ION Anca Veronica
"Gh. Mihoc - C. Iacob" Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania
Title: Qualitative and numerical study of a system of delay differential equations modeling leukemia (details)
This work is a review of some previous works of ours concerning the investigation of a system of two delay differential equations, system that models the periodic chronic myelogenous leukemia. The system consists of two equations, one for the proliferating cells and one for the so-called resting cells, and depends on five parameters. The equation for resting cells is independent of the other one. For this equation we study the stability of the two equilibria, the Hopf bifurcation, and the Bautin bifurcation. Then we study the behavior of the proliferating cells (that is determined by that of the resting cells).
ION Stelian
Institute of Statistical Mathematics and Applied Mathematics of Romanian Academy, Romania
Title: Water flow on vegetated hill. Shallow water equations model (details)
The plant presence on hills creates a resistance force to the water flow and influences the process of water accumulation on the soil surface. The large diversity of the growing plants on a hill makes the elaboration of an unitary model of water flow over a soil covered by plants very difficult. At a hydrographic basin scale, there are variations in the geometrical properties of the terrain (curvature, orientation, slope) and vegetation density or vegetation type etc. A mathematical model for the water flow on a hill covered by variable distributed vegetation is proposed in this article. The space averaging method is used to define an unique continuous model associated to a heterogeneous fluid-soil-plant mechanical system. Some mathematical properties of the model are presented and the behavior of a simplified model is illustrated by numerical results.
Universitatea de Vest din Timisoara, Romania
Title: Dynamical analysis of a fractional-order Hindmarsh-Rose model (details)
This work is dedicated to the stability and bifurcation analysis of a model of neuronal activity of Hindmarsh-Rose type and of fractional order. First, a two-dimensional model is considered, with respect to the membrane potential in the axon and the transport rate of sodium and potassium ions through fast ion channels. This model is later complemented by the addition of a third-order fractional differential equation that takes into account a slow adaptation current. The main purpose of this paper is to demonstrate that existing mathematical models of neuronal activity can be improved by using fractional derivatives instead of classical integer-order derivatives and that these new models reflect a better understanding of the biological reality, as suggested by the experimental results. Results of numerical simulations are also presented to validate the theoretical results.
KOHR Mirela
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania
Title: Boundary value problems of transmission type for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in weighted Sobolev spaces (details)
The purpose of this talk is to present existence and uniqueness results in $L^2$-weighted Sobolev spaces for transmission problems for the nonlinear Darcy-Forchheimer-Brinkman and Navier-Stokes systems in two complementary Lipschitz domains in ${mathbb R}^3$. We exploit layer potential theoretic methods for the Stokes and Brinkman systems combined with fixed point theorems in order to show the desired existence and uniqueness results. Note that the Brinkman-Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows. This talk is based on joint work with Massimo Lanza de Cristoforis (Padova), Sergey E. Mikhailov (London) and Wolfgang L. Wendland (Stuttgart).
LITCANU Gabriela
Institute of Mathematics "O. Mayer", Romania
Title: About patterns driven by chemotaxis (details)
The pattern formation is a key process in the development of living systems. It describes the interplays between members of species at the intercellular or intracellular levels. We consider a coupled chemotaxis-haptotaxis system which describes a large variety of biological or medical phenomena. We investigate how both the change of parameters of the system and the singularities of the chemosensitivity term can generate pattern formation.
University of Poitiers, France
Title: Some generalizations of the Cahn-Hilliard equation (details)
Our aim in this talk is to discuss some variants of the Cahn-Hilliard equation. Such models have, in particular, applications in biology and image inpainting.
NEAMTU Mihaela
West University of Timisoara, Romania
Title: Hopf bifurcation analysis for the model of the hypothalamic-pituitary-adrenal axis with distributed time delay (details)
In the present paper we analyze the mathematical model of the hypothalamus-pituitary-adrenal axis. Since there is a spatial separation between the brain, where the hypothalamus and pituitary are situated, and the kidney, where the adrenal glands are situated, time is needed for transportation of the hormones between the glands. Thus, the distributed time delays are considered as both weak and Dirac kernels. The model, described by a nonlinear differential system with distributed time delay, is analyzed regarding the stability and bifurcation behavior. The last part contains some numerical simulations to illustrate the effectiveness of our results. Moreover, the behavior of the fractional differential time delay model is simulated.
OPREA Iuliana
Colorado State University, USA
Title: Spatiotemporal compex dynamics in anisotropic fluids (details)
Spatially extended systems driven far from equilibrium may exhibit spatiotemporal complex dynamics manifested through spatiotemporal chaos, intermittency, defects, phase turbulence, etc. We present a comprehensive theoretical framework for the classification and characterization of the spatiotemporal complex dynamics in one of the most challenging pattern forming anisotropic systems, the electroconvection of nematic liquid crystals, based on Ginzburg Landau type amplitude equations.
SIMION STOILOW Institute of Mathematics, Romania
Title: Saffman-Taylor instability for a non-Newtonian Fluid (details)
We study the linear stability of a steady displacement of an Oldroyd-B fluid by air in a Hele-Shaw cell. We perform a depth-average procedure, across the Hele-Shaw gap, in the dynamic boundary condition at the interface. The new element is an exact formula of the growth rate of perturbations, obtained for small $Deborah$ numbers. When the $Deborah$ numbers are equal, our growth rate is quite similar to the Saffman-Taylor formula for a Newtonian liquid displaced by air. We prove the non-Newtonian destabilizing effects.
University Politehnica of Bucharest, Romania
Title: Elastoplastic models with continuously distributed defects: dislocations and disclinations, for finite and small strains. (details)
The models describe the behaviour of crystalline materials with microstructural defects. The mathematical measures of defects are defined in terms of the incompatible tensor fields and are based on anholonomic configurations. The non-local, diffusion-like evolution equations describe the coupling between dislocations and disclinations. The numerical results emphasize the effects induced by the initial heterogeneity in distribution of disclinations and by the two diffusion like parameters, involved in the evolution equations.
PETCU Madalina
University of Poitiers, France
Title: Parallel matrix function evaluation via initial value ODE modelling (details)
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions f(A), a question of major interest in the numerical linear algebra. To this end, we model f(A) as the solution at a finite time T of a time dependent equation. We use parallel algorithms, such as the parareal method, on the time interval [0; T] in order to solve the evolution equation obtained.
Institutul de Matematica "S.Stoilow" al Academiei Romane, Romania
Title: The flow through fractured porous media along Beavers-Joseph interfaces (details)
We study the flow through a porous medium fractured in blocks by an $varepsilon$-periodic distribution of fissures filled with a Stokes fluid. We assume that the filtration fluid through the blocks is obeying the Darcy's law and that it is coupled with the Stokes fluid from the fissures by a Beavers-Joseph type interface condition. We prove the existence and uniqueness of this coupled flow when it is confined by an impermeable boundary. The measure of the interface being of order $1/varepsilon$ as $varepsilon rightarrow 0$, we study the asymptotic behavior of the flow in the case when the permeability is of unity order and the Beavers-Joseph transfer coefficient is of $varepsilon$-order. We prove that the two-scale limits of the velocities and of the pressures verify a well-posed problem, from which the so-called local problems can be decoupled and, therefore, the macroscopic(homogenized) behavior is explicit.
POPA Constantin
Ovidius University of Constanta, Romania
Title: On Single Projection Kaczmarz Extended-Type Algorithms (details)
The Kaczmarz Extended (KE) algorithm has been proposed by the author in cite{ref2, ref3}, as an extension of the Kaczmarz-Tanabe algorithm from cite{ref4}, to inconsistent linear least squares problems. It uses in each iteration orthogonal projections onto the hyperplanes determined by all the rows and all the columns of the system matrix. Recently, in the paper cite{ref5}, the authors proposed a single projection KE – type algorithm, which in each iteration uses orthogonal projections onto the hyperplanes determined by only one row and one column. If the projection row and column indices $i$ and $j$ are selected {it at random with probability proportional with a certain quotient of the norm of the $i$-th row, and $j$-th column}, respectively, they prove that the sequence of approximations so generated converges in Expectation to a least square solution of the problem. In this paper we propose two single projection KE – type algorithms, in which the projection indices are selected in an {it almost cyclic}, and {it remote control} maner, respectively (see e.g. cite{ref1}). We prove that the sequence of approximations generated in each case converges in norm to a least square solution of the problem. Numerical experiments and comparisons are also presented. begin{thebibliography}{10} bibitem{ref1} Y. Censor, A.Z. Stavros, textit{Parallel optimization: theory, algorithms and applications}. ''Numer. Math. and Sci. Comp.'' Series, Oxford Univ. Press, New York, 1997. bibitem{ref2} C. Popa, textit{Least-squares solution of overdetermined inconsistent linear systems using Kaczmarz's relaxation}. Intern. J. Comp. Math., {bf 55} (1995), pages 79-89. bibitem{ref3} C. Popa, textit{Extensions of block-projections methods with relaxation parameters to inconsistent and rank-defficient least-squares problems}. {it B I T}, {bf 38(1)} (1998), pages 151-176. bibitem{ref4} K. Tanabe, textit{ Projection Method for Solving a Singular System of Linear Equations and its Applications}, Numer. Math., {bf 17} (1971), pages 203-214. bibitem{ref5} A. Zouzias A., N.M. Freris, textit{ Randomized Extended Kaczmarz for Solving Least Squares}, arXiv:1205.5770v3 (5.01.2013). end{thebibliography}
United States Naval Academy, USA and Romania
Title: Border-Collision Bifurcations in A Piece-Wise Smooth Planar Dynamical System Associated with Cardiac Potential (details)
The talk addresses the bifurcations of a two-dimensional non-linear dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation. The dynamical behavior of this continuous (but only piece-wise smooth) model transitions from simple (a unique attracting cycle) to complicated (co-existence of stable cycles) as the stimulus period is decreased from large towards zero. The first bifurcation, of discontinuous period-doubling type, results from the collision of two cycles with a switching manifold. For stimuli periods just shorter than collision time, of the two cycles about to collide, the 2:1 escalator is stable and the alternans solution is unstable; with those co-exists a stable 1-escalator whose orbit lays away from the switching manifold. The principal results show that the dynamical system associated with the collision exhibits two distinct types of domains of attraction, some impossible in smooth dynamics.
Babes-Bolyai University of Cluj-Napoca, Romania
Title: Mathematical models of stem cell transplantation (details)
We present simple mathematical models expressed as three-dimensional ordinary differential systems for describing the dynamics of three cell lines after allogeneic and autologous stem cell transplantation [1], [3]. The evolution ultimately leads either to the normal hematopoietic state achieved by the expansion of normal cells and the elimination of the cancer cells, or to the leukemic hematopoietic state characterized by the proliferation of the cancer line and the suppression of the normal cells. A theoretical basis for the control of post-transplant evolution is provided for the allogeneic transplant [4]. Thus, we describe several scenarios of change of system parameters by which a bad post-transplant evolution can be corrected and turned into a good one, and we propose therapy planning algorithms for guiding the correction treatment [2]. We also conclude about the effectiveness of the autologous transplantation as therapeutic procedure for AML [3]. References: [1] R. Precup et al., Mathematical modeling of cell dynamics after allogeneic bone marrow transplantation, Int. J. Biomath. 5 (2012), 1250026, 1--18. [2] R. Precup et al., A planning algorithm for correction therapies after allogeneic stem cell transplantation, J. Math. Model. Algor. 11 (2012), 309--323. [3] R. Precup, Mathematical understanding of the autologous stem cell transplantation, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 10 (2012), 155--167. [4] R. Precup, M-A Serban, D Trif, Asymptotic stability for a model of cell dynamics after allogeneic bone marrow transplantation, Nonlinear Dyn. Syst. Theory 13 (2013), 79--92.
Astronomical Institute of the Romanian Academy, Romania
Title: Modelling of pulsations of giant stars (details)
We tackle the problem of interaction between convection and pulsations in giant stars. We present the results of numerical computations of oscillation properties of a model of the G9.5 giant $\epsilon$ Oph, based on a new treatment of convection. The effects on modes stability and modes inertia are pointed out.
State University of New York at New Paltz, United States
Title: Dynamic Networks: From Connectivity to Temporal Behavior (details)
Many natural systems are organized as networks, in which the nodes (be they cells, individuals or populations) interact in a time-dependent fashion. We illustrate how the hardwired structure (adjacency graph) can affect dynamics (temporal behavior) for two particular types of networks: one with discrete and one with continuous temporal updates. The nodes are coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We develop new measures (such as probabilistic bifurcations and extensions of Julia sets) to compare the effects of different ways of increasing connectivity: by altering edge weights versus edge density versus edge configuration. We determine that the adjacency spectrum is a poor predictor of dynamics, that increasing the number of connections is not equivalent to strengthening them, and that there is no single factor among those we tested that governs the stability of the system. We discuss the potential applications of our results towards increasing our understanding of neural dynamics and genetic replication processes.
University Politehnica of Bucharest, Romania
Title: Optimal control of Imatinib treatment in a competition model of Chronic Myelogenous Leukemia with immune response (details)
In this work, we present an optimal control strategy for a delay differential system with application to the drug therapy of Chronic Myelogenous Leukemia. The mathematical model is represented by a nonlinear system of five equations from population dynamics which is based on the competition between normal cells and leukemic cells (stem-like and mature). The effect of the immune system through anti-leukemia T-cells is also included in the model. The effect of Imatinib therapy is introduced in the form of several factors controlling cell multiplication and mortality for leukemia cell populations and the evolution of immune cell population. The optimal control method is applied to eliminate the leukemic cells whilst minimizing the amount of drug. In order to derive necessary optimality conditions in the form of Pontryagin’s minimum principle, discretization methods are applied and the delayed control problem is augmented to a nondelayed problem to which Pontryagin’s minimum principle is applicable. The obtained results are illustrated by numerical simulations and discussed in view of the optimal treatment approach.
STRUGARU Magdalena
BCAM, Spain
Title: Simulation of necking phenomenon in a polyconvex material (details)
Necking is a kind of deformation that typically undergo polymers and ductile metals when subjected to extreme tensile extension. It is a plastic deformation (which still can be modelled by elasticity theory) consisting of the decrease of cross-sectional area in a small region (typically, the center) of the material. Not all materials present necking, and in fact, it is at first glance counter-intuitive that necking needs less energy than the homogeneous deformation satisfying the same boundary conditions. Most of the available literature deals with general sufficient conditions for which necking does not occur or with one-dimensional models for which necking does occur. The goal of this work is to show analitically and numerically that necking is possible in some materials whose energy density is compatible with the existence theorems in nonlinear three-dimensional elasticity. We aim at finding an isotropic polyconvex function for which the homogeneous deformation (which is always a stationary point) is not a minimizer of the energy. This is not an easy task because polyconvexity implies at once that the homogeneous deformation satisfies all usual necessary conditions for local miminizers. Then, for that instance of energy density, we compute numerically the global minimizer of the energy. We present two classes of experiments: one that computes the minimizer using a trust-region Newton method and one that solves the minimization problem using an augmented Lagrangian algorithm.
SURAN Marian Doru
Astronomical Institute of the Romanian Academy, Romania
Title: Exploring the Space of Stellar Parameters for PLATO2 Space Mission Targets Using CESAM2k and LNAWENR/ROMOSC Codes (details)
In order to extract the basic stellar parameters we use the asteroseismic inversion method where the observed oscillation frequencies are used to estimate the stellar parameters. The inversion shall be understood such that the best estimated parameters for a given star correspond to the model input parameters for the model that shows frequencies most similar to the observed ones. We have computed a wide grid of stellar models and their associated oscillation frequencies and we have designed a tool to evaluate the value of $\chi ^{2}$ on that grid for different possible sets of observational data. Preliminary results were been obtained for some observed CoRoT and Kepler missions targets.
Purdue University Calumet, USA
Title: A hybrid mathematical model for cell motility in angiogenesis (details)
The process of angiogenesis is regulated by the interactions between various cell types such as endothelial cells and macrophages, and by biochemical factors. In this talk, we present a hybrid mathematical model in which cells are treated as discrete units in a continuum field of a chemoattractant that evolves according to a system of reaction-diffusion equations, whereas the discrete cells serve as sources/sinks in this continuum field. It incorporates a realistic model for signal transduction and VEGF production and release, and gives insights into the aggregation patterns and the factors that influence stream formation. The model allows us to explore how changes in the microscopic rules by which cells determine their direction and duration of movement affect macroscopic formations. In particular, it serves as a tool for investigating tumor-vessel signaling and the role of mechano-chemical interactions of the cells with the substratum.