For Fall 2016 our group seminar will take place each Friday from 3:00pm to 4:00pm in JO 3.516

 Speaker Title/Abstract/Affiliation Date Time Room Host Wieslaw Krawcewicz Multiple Symmetric Solutions to Newtonian Systems with Symmetries: The Gradient Equivariant Degree Approach Finding $p$-periodic solutions to the second order Newtonian system of the type $$\tag{1}\label{eq:sys1} \ddot x=-\nabla f(x),$$ where $f:\mathbb{R}^n\to \mathbb{R}$ is a $C^2$-differentiable function, can be reduced to the problem of finding critical points of the functional $J: \mathscr H\to \mathbb{R}$, where $\mathscr H:=H^1(S_p^1;V)$, $V:=\mathbb{R}^n$ and $S_p^1:=\mathbb{R}/(p/2\pi) \mathbb{Z}$, is given by $$J(x):=\int_0^p \left(\frac 12 |\dot x(t)|^2-f(x(t)) \right) dt, \quad |x(t)|^2:=x(t)\bullet x(t).$$ It is well known that the functional $J$ is $S^1$-invariant (where $S^1$ acts on $\mathscr H$ by shifting the argument) (see for example [7,4] see also [8,9,10]), however the system \eqref{eq:sys1} is time reversible, which induces a natural $O(2)$-action on $\mathscr H$. In the case $V$ is an orthogonal $\Gamma$-representation (we assume that $\Gamma$ is a finite group acting on $V=\mathbb{R}^n$ by permuting vector-coordinates), the problem \eqref{eq:sys1} can be reduced to the $G$-equivariant equation, $G=\Gamma\times O(2)$, $$\nabla J(x)=0,\quad x\in \mathscr H,$$ and the gradient $G$-equivariant degree (developed by Geba in [5]) can be applied. In my talk, I will present our most recent results (obtained together with Mietek Dabkowski and Yanli Lu, cf. [6]) on the usage of the gradient $\Gamma\times O(2)$-equivariant degree to classify according to their symmetries the orbits of non-constant $p$-periodic solutions to system \eqref{eq:sys1}, including the properties of the Euler ring $U(\Gamma\times O(2))$ (see [11,3]), computations of the $\Gamma\times O(2)$-basic degrees (see [2,1]), etc. References: [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree. AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. [2] Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App. 8 (2010), 1-74. [3] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2) \times S^1$-symmetric variational problems: Equivariant gradient degree approach.Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, June 18-24, 2008, AMS Contemporary Mathematics, 540 (2010), 45-84. (2008) [4] J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns 218 (2005), 216-252. [5] K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis II (Frascati, 1995), Progr. Nonlinear Differential Equations App. 27, Birkh\"auser, Boston, 1997, 247-272. [6] M. Dabkowski, W. Krawcewicz and Y. Lv, Multiple Periodic Solutions for Symmetric Second Order Newtonian Systems with Even Potentials, Preprint 2015. [7] H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal. 68 (2008), 1479-1516. [8] S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory. Nonlinear Anal. 23 (1994), 83-102. [9] S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries. Topol. Methods Nonlinear Anal. 9 (1997), 383-417. [10] S. Rybicki, Degree for $S^1$-equivariant strongly indefinite functionals. Nonlinear Anal. 43 (2001), 1001-1017. [11] T. tom Dieck, Transformation Groups. Walter de Gruyter, 1987.   Department of Mathematical Sciences, University of Texas at Dallas Jan 23, 2015 3pm-4pm FO 2.404 Jameson Graber Mean field games: an introduction Mean field game theory has been making many advances in the past decade. Its many applications are found in economics and finance, networks and cybersecurity, and even biology. In this presentation we introduce the fundamentals of the theory, starting with an explanation of the two conceptual components--game theory and mean field theory--and then putting them together. We will see how this leads to interesting mathematical models composed of nonlinear partial differential equations, and we will discuss some of the technical tools used to analyze them. Finally, we will present some of latest results on mean field games and make some remarks on open problems.   Department of Management, University of Texas at Dallas Jan 30, 2015 3pm-4pm FO 2.404 Robert Gregg Energy Shaping Approaches from Robot Walking to Lower-Limb Orthotics Department of Bioengineering, University of Texas at Dallas Feb 6, 2015 3pm-4pm FO 2.404 Ivan Gudoshnikov On the stability of perturbed linear dynamical systems in ordered spaces Department of Mathematical Sciences, University of Texas at Dallas Feb 13, 2015 3pm-4pm FO 2.404 Guojun Gan Math Colloquium Department of Mathematics, University of Connecticut Feb 20, 2015 2pm- 3pm FO 2.404 Viswanath Ramakrishna rescheduled to April 10 due to inclement weather Feb 27, 2015 3pm-4pm FN 2.106 Hasan Poonawala Preserving Strong Connectivity in Communication Networks Subtitle: Use of the structure of the Perron eigenvector for reducible stochastic matrices in control applications Abstract: Co-operation between multiple mobile robots often requires that the communication network formed by these robots be strongly connected. As the robots move in order to achieve some task, the loss of communication links may lead to loss of strong connectivity. We model the directed communication network using a stochastic matrix whose entries depend on the states of the robots. When the network is strongly connected, the stochastic matrix is irreducible and its Perron eigenvector has strictly positive components. If the network loses strong connectivity due to motion of the robots, the stochastic matrix may become reducible and some of the components of its Perron eigenvector may vanish. In this talk, we will describe which components will vanish, and how this knowledge can be used to derive a control law which preserves the strong connectivity of the network, under suitable conditions. Department of Engineering & Computer Science, University of Texas at Dallas Mar 6, 2015 3pm-4pm FO 2.404 Anne Martin Improving Amputee Walking Gait using Ideas from Underactuated Robot Control Because current commercial prostheses do not completely replicate the function of the physiological foot and ankle, amputee walking gait is typically both less efficient and less stable than healthy human walking. Powered prostheses offer one potential solution, although current devices and control methods are unable to completely restore healthy gait. In addition, for both powered and passive prostheses, the effects of foot design on walking ability are largely unknown. Thus, the development of a modeling and control method that can predict both healthy and amputee walking could allow for improvements in the physical design of prostheses and in the control design of powered prostheses. Unfortunately, existing models are either too computationally expensive or too simplistic to be used for this purpose. This talk presents a model of intermediate complexity based on the Hybrid Zero Dynamics (HZD) control approach that was originally developed for point-foot, underactuated robots. By incorporating feet into the model and correctly choosing the objective function for gait optimizations, the HZD-based model can be extended to accurately predict the lower-limb kinematics and energy expenditure of healthy human walking across a wide range of speeds. To use the model to investigate human amputee gait, it has been further extended to allow asymmetrical gait. This talk will describe the model, discuss the technical details of its extension from the original robotic formulation, demonstrate its predictive capabilities for healthy human walking gait, and present ongoing work in predicting human amputee gait and in developing a control formulation for a powered prosthesis. Department of Mechanical Engineering, University of Texas at Dallas Mar 13, 2015 3pm-4pm FO 2.404 Spring break Mar 20, 2015 Martin Brokate Weak differentiability of scalar hysteresis operators Technische Universitaet, Munich Mar 25, 2015 3pm-4pm FO 2.208 Mohamed Khamsi On fixed Point Theory of Monotone Mappings Nonexpansive mappings are those maps which have Lipschitz constant equal to 1. The fixed point theory for such mappings is rich and varied. It finds many applications in nonlinear functional analysis. The existence of fixed points for nonexpansive mappings in Banach and metric spaces have been investigated since the early 1960s. Recently a new direction has been discovered dealing with the extension of the Banach Contraction Principle to metric spaces endowed with a partial order. The first attempt was successfully carried by Ran and Reurings. In particular, they showed how this extension is useful when dealing with some special matrix equations. Another similar approach was carried by Nieto et al. and used such arguments in solving some differential equations. Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order. In this talk, we will discuss the case of monotone nonexpansive mappings. Some of the results obtained are new and open the door to some new directions in metric fixed point theory. Department of Mathematical Sciences, University of Texas at El Paso Mar 27, 2015 3pm-4pm FO 2.404 Viswanath Ramakrishna The Controlled Invariance Problem Department of Mathematical Sciences, University of Texas at Dallas Apr 10, 2015 2pm-3pm FO 2.404 Masoud Yari Nonlinear analysis of a mathematical model of propagation and degradation of vegetation patterns in semi-arid ecosystems online presentation via WebEx The purpose of this talk is to present and analyze a macro-scale mathematical model of vegetation propagation/degradation in semi-arid ecosystems. By using a non-linear analysis, I will derive dynamic phase transition equations and introduce the phase transition parameters. These phase transition parameters play a fundamental role in the structural stability analysis of solutions and geometric objects in the phase space as well as understanding transitions to higher states. I will focus on different localized patterns and their transitions, following by a discussion on strategies in qualitative parameter approximation for such systems. Such an analysis is useful in determining uncertainty regions, finding tipping points, and revealing some other hidden dynamics. The idea behind this approach goes back to pioneer studies of morphogenesis in biological systems. The principle of local self-activation and lateral inhibition, in the study of self-assembly of patterns in morphogenesis, is well formulated in works of Gierer and Meinhardt. The same principle can model the dynamics in semi-arid ecosystems that exhibit localized patterns in vegetation due to limitation of water resources, leading to formation of visually captivating features in aerial images, such as spot, gap, and labyrinth patterns as well as roll patterns on gradual slopes. School of Engineering & Computing Sciences, Texas A&M University-Corpus Christi Apr 17, 2015 3pm-4pm FO 2.404 Zhaosheng Feng Degenerate Parabolic System and Its Approximate Solutions In this talk, we are concerned with approximate solutions to a degenerate parabolic system. We provide a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the degenerate parabolic system, a partial differential equation that is the dispersion model of biological populations with both density-dependent diffusion and nonlinear rate of growth. We present the integral forms of the Abel equation with the initial condition. By virtue of the integral forms and the Banach Contraction Mapping Principle we derive the asymptotic expansion of bounded solutions in the Banach space, and use the asymptotic formula to construct approximate solutions to the degenerate parabolic system. Department of Mathematics, University of Texas-Pan American Apr 24, 2015 3pm-4pm FO 2.404 Sophia Jang Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton There are many toxin producing phytoplankton (TPP) species in natural systems that can have extremely harmful effects on the pop- ulations interacting with them. We rst review several systems of plankton interactions with TPP consideration in the literature. We then present our phytoplankton-zooplankton models with toxin pro- ducing phytoplankton to study the effects of TPP upon extinction and persistence of the populations. One model assumes spatial ho- mogeneity while the other model considers spatial heterogeneity by incorporating diffusion. It is concluded that TPP may promote sur- vival of phytoplankton population and may destabilize the interactions depending on parameter regimes. Numerical simulations indicate that passive diffusion of both populations can simplify the dynamics of the interactions and exhibit plankton patchiness. Department of Mathematics and Statistics Texas Tech University May 1, 2015 3pm-4pm FO 2.404 Oleg Makarenkov Bifurcation of stable limit cycles in switched systems Department of Mathematical Sciences, University of Texas at Dallas May 8, 2015 3pm-4pm FO 2.404
 Speaker Title/Abstract/Affiliation Date Time Room Slides Host Dmitry Rachinskiy Combining plays: from Prandtl-Ishlinskii operator to network models Sep 5, 2014 3pm-4pm FO 2.404 Maxim Arnold Escaping orbits in switching Hamiltonian systems Sep 12, 2014 4pm- 5pm FO 2.404 Maxim Arnold Escaping orbits in switching Hamiltonian systems II Sep 19, 2014 3pm-3:40pm FO 2.404 Faculty meeting Sep 26, 2014 Zalman Balanov Complex structures in Algebra, Analysis, Topology and Differential Equations Oct 3, 2014 3pm-4pm FO 2.404 Yifei Lou An introduction to mathematical models in image processing with a focus on PDEs and optimization techniques Oct 10, 2014 3pm-4pm FO 2.404 Sergey Antonyan The Gromov-Hausdorff metric and hyperspaces The Gromov-Hausdorff distance $d_{GH}$ is a useful tool for studying topological properties of families of metric spaces. M. Gromov first introduced the notion of the Gromov-Hausdorff distance $d_{GH}$ in his ICM 1979 address in Helsinki on synthetic Riemannian geometry. Two years later $d_{GH}$ appeared in the book M. Gromov. Structures metriques pour les varietes riemanniennes. Vol. 1 of Textes Mathematiques [Mathematical Texts], CEDIC, Paris, 1981. Edited by J. Lafontaine and P. Pansu. For two compact metric spaces $X$ and $Y$ the number $d_{GH}(X,Y)$ is defined to be the infimum of all Hausdorff distances $d_H(i(X),j(Y))$ for all metric spaces $M$ and all isometric embeddings $i:X\to M$ and $j:Y\to M.$ Clearly, the Gromov-Hausdorff distance between isometric spaces is zero; it is a metric on the family $GH$ of isometry classes of compact metric spaces. The metric space $(GH,d_{GH})$ is called the Gromov-Hausdorff hyperspace. It is a challenging open problem to understand the topological structure of this metric space. This talk contributes towards this problem. We mainly are interested in the following subspaces of $GH$ denoted by $GH(\mathbb{R}^n),$ $n\ge 1$, and called the Gromov-Hausdorff hyperspace of $\mathbb{R}^n.$ Here $GH(R^n)$ is the subspace of $GH$ consisting of the classes $[E]\in GH$ whose representative $E$ is a metric subspace of the Euclidean space $\mathbb{R}^n.$ One of the results in this talk asserts that $GH$ is homeomorphic to the orbit space $2^{R^n}/ E(n),$ where $2^{R^n}$ is the hyperspace of all nonempty compact subsets of $\mathbb{R}^n$ endowed with the Hausdorff metric and $E(n)$ is the isometry group of $\mathbb{R}^n.$ This is applied to prove that $GH(\mathbb{R}^n)$ is homeomorphic to the Hilbert cube with a removed point.   Department of Mathematics, National Autonomous University of Mexico Oct 17, 2014 3pm-4pm FO 2.404 Wieslaw Qingwen Hu A state-dependent model with unimodal feedback Oct 24, 2014 3pm-4pm FO 2.404 Vadim Azhmyakov Modern Directions in Optimization and Control of Switched Systems: New Challenges for Mathematicians Hybrid and switched systems are mathematical models of heterogeneous control processes consisting of a continuous part, a finite number of continuous controllers, and a discrete supervisor. These models can represent an extremely wide range of systems of practical interest and are accepted as realistic models, for instance, in industrial electronics, power systems engineering, maneuvering aircrafts, automotive control systems, controllable chemical processes and communication networks. The emergence of a discrete event systems modeling framework is providing a new perspective for some important modern control processes and also constitutes a new challenges for Applied Mathematicians. In our talk, we discuss some specific families of switched systems, and the corresponding dynamic optimization problems. The class of problems to be considered concerns models, where discrete transitions are being triggered by the continuous dynamics, and are accompanied by discontinuous changes in the continuous state variable (state jumps). The control objective is to minimize a cost functional, where the control parameters are not only usual control inputs but also the discrete events associated with the switched nature of the systems dynamics. We describe some newly elaborated implementable solution procedures, consider illustrative examples and point the main directions of the recent investigations in the area.   Faculty of Electronic and Biomedical Engineering, University of Antonio Nari�o Neiva, Republic of Colombia Oct 31, 2014 3pm-4pm FO 2.404 Oleg Vladimir Dragovic Introduction to pseudo-integrable billiards Nov 7, 2014 3pm-4pm FO 2.404 Mieczyslaw Dabkowski Catalan States of Lattice Crossing Nov 14, 2014 3:15pm- 4:15pm FO 2.404 Byungik Kahng homepage Trichotomy of Singularities of 2-Dimensional Bounded Invertible Piecewise Isometric Dynamics It is known that the singularities of 2-dimensional bounded invertible piecewise isometric dynamical systems can be classified as, removable, shuffling and sliding singularities, based upon their geometrical traits; and that only the sliding singularity can generate Devaney-chaos, while the others remain innocuous. However, the afore-mentioned classification had been somewhat incomplete in that the clear distinction between the sliding and the shuffling singularities had been unavailable. The presented research resolves this difficulty and completes the trichotomy, by completely characterizing the nature of sliding singularity. We use the invertible dual rotation map (or the Goetz map), as the main tool. The above abstract may change. I am assuming that I can complete what I am working on by the end of October. If I do not, then I cannot claim that my research would, "complete the trichotomy." I may end up talking about the background only and leave the technical part for next talk several weeks or months later.   Department of Mathematics and Information Sciences, University of North Texas at Dallas Nov 21, 2014 3pm-4pm FO 2.404 Oleg Thanksgiving holiday Nov 28, 2014 Julian Newman homepage Math Colloquium Dec 5, 2014 3:00pm-4:00pm FO 2.404