Seminar on Differential Equations and Nonlinear Analysis

 

Organizers: Alessandro Arsie (alessandro dot arsie at utoledo dot edu)

                     Chunhua Shan (chunhua dot shan at utoledo dot edu)

Time: Tuesday, 4:00-5:00PM

Location: UH 2210

 

All are welcome to attend.

If you would like to present your work or an interesting paper you have read, please contact me or Dr. Arsie.

If you have any suggestions on this seminar, please also let us know. Thanks.


 

Spring Semester, 2018

 

Talk:  Relativistic Treatment of Confined Hydrogen Atoms via Numerical Approximations

Speaker: Jacob Noon, The University of Toledo

Date: 4:00-5:00PM, Tuesday, April 10, 2018

Abstract: The study of particles and atoms confined to spherically symmetric regions have been used to illustrate the differences between classical and quantum systems since Erwin Schrodinger’s famous equation was published in 1926. Paul Dirac later added his own equation, the Dirac Equation (1928), which integrated the quantum principles that had been developing in the decades prior to Schrodinger’s equation with the principles of Einstein’s Special Theory of Relativity.


The literature on Hydrogen atoms confined to spherically symmetric regions (using the Schrodinger equation) is abundant, with consensus on the results. To the best of our knowledge there exists no relativistic treatment to this problem (i.e., using the Dirac equation). Some reasons for this are the complexity of the Dirac equation, its solutions, and problems that arise when trying to satisfy the boundary conditions.

In this talk, I will present solutions to the given problem, as well as limitations that are mathematical and physical in nature. The methods used to obtain solutions involve solving systems of first order linear ordinary differential equations analytically, and computing the roots of ratios of Kummer functions via two different numerical methods.

 

 

Talk:  A revisit to the Jordan canonical form for a complex square matrix

Speaker: Dr. Biao Ou, The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 13, 2018

Abstract: I will first look at the canonical form of two by two and three by three matrices via a rank one matrix. Next, I look at the process in which we see the Jordan canonical form by first applying a much easier Schur's theorem and then considering a system of linear differential equations. I will also apply the Jordan canonical form to a sequence of numbers satisfying a linear iterative equation.

 

 

Talk:  Factorization of differential operators on the algebra of densities on the line (II)

Speaker: Dr. Ekaterina Shemyakova, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 20, 2018

Abstract: I shall speak about my new work where I explore factorization problem for differential operators on the algebra of densities. This work is a starting point for one of the directions of my research program, namely study of differential operators acting on geometric quantities and their Darboux transformations in classical and super setting. 

I shall recall classical facts concerning factorization of differential operators and then introduce the algebra of densities and differential operators on this algebra. In particular, I shall give a new motivation for the introduction of the algebra of densities basing on the work of Duval-Ovsienko'96. Note that the original motivation of Khudaverdian-Voronov'02 was the geometry of Batalin-Vilkovisky quantization.

I shall show that for differential operators on the algebra of densities on the line (unlike the familiar setting) there are obstructions for factorization.  I shall analyze these obstructions.  In particular, for the "generalized Sturm–Liouville" operators acting on the algebra of densities on the line, I shall show our criterion of factorizabily in terms of solution of the classical Sturm–Liouville equation.  I shall also show the possibility of an incomplete factorization.

 

Talk:  Factorization of differential operators on the algebra of densities on the line (I)

Speaker: Dr. Ekaterina Shemyakova, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 13, 2018

Abstract: I shall speak about my new work where I explore factorization problem for differential operators on the algebra of densities. This work is a starting point for one of the directions of my research program, namely study of differential operators acting on geometric quantities and their Darboux transformations in classical and super setting. 

I shall recall classical facts concerning factorization of differential operators and then introduce the algebra of densities and differential operators on this algebra. In particular, I shall give a new motivation for the introduction of the algebra of densities basing on the work of Duval-Ovsienko'96. Note that the original motivation of Khudaverdian-Voronov'02 was the geometry of Batalin-Vilkovisky quantization.

I shall show that for differential operators on the algebra of densities on the line (unlike the familiar setting) there are obstructions for factorization.  I shall analyze these obstructions.  In particular, for the "generalized Sturm–Liouville" operators acting on the algebra of densities on the line, I shall show our criterion of factorizabily in terms of solution of the classical Sturm–Liouville equation.  I shall also show the possibility of an incomplete factorization.

 

 

Talk:  Flat $F$-manifolds, Miura invariants and integrable systems of conservation laws

Speaker: Dr. Alessandro Arsie, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 6, 2018

Abstract: In this talk, I will present the extension to the case of systems of integrable conservation laws of some of the results proved for scalar equations in Arsie, Moro, Lorenzoni (Integrable viscous conservation laws, Nonlinearity 2015) and in Arsie, Moro, Lorenzoni (On Integrable Conservation Laws, Proceedings of the Royal Society A, 2014). 

For such systems, I will show that the eigenvalues of a matrix obtained from the quasilinear part of the system are invariants under Miura transformations, and I will highlight how these invariants are related to dispersion relations. Furthermore, focusing on one-parameter families of dispersionless systems of integrable conservation laws associated to the Coxeter groups of rank $2$ found in Arsie, Lorenzoni (Complex reflection groups, logarithmic connections and bi-flat F-manifolds, Letters in Math. Physics 2017), I will discuss the corresponding integrable deformations up to order $2$ in the deformation parameter $\epsilon$. 

Each family contains both bi-Hamiltonian and non-Hamiltonian systems of conservation laws and therefore we use it to probe to which extent the properties of the dispersionless limit impact the nature and the existence of integrable deformations. It turns out that besides two values of the parameter, all deformations at order one in $\epsilon$ are Miura-trivial, while all those of order two in $\epsilon$ are essentially parameterized by two arbitrary functions of single variables (the Riemann invariants) both in the bi-Hamiltonian and in the non-Hamiltonian case.

In the two remaining cases (the two special values of the parameter), due to the existence of non-trivial first order deformations, there is an additional functional parameter. These are the results of a recent joint work with Paolo Lorenzoni (Universita' di Milano-Bicocca), to appear in Journal of Integrable Systems (Oxford University Press).

 

 

 

 

Fall Semester, 2017

 

Talk:  Lie symmetries of the canonical Lie group connection

Speaker:  Professor Gerard Thompson, The University of Toledo

Date: 4:00-5:00PM, Thursday, December 5, 2017

Abstract: It is well known that any Lie group carries a canonical symmetric although usually not metric linear connection - Cartan's so-called "0"-connection. In this talk we investigate Lie symmetries of the canonical connection. We shall focus particularly on the codimension one abelian nilradical case for which many symmetries and first integrals may be written down explicitly.

 

 

Talk:  Hopf bifurcation of planar systems

Speaker:  Chanaka Kottegoda, The University of Toledo

Date: 4:00-5:00PM, Thursday, November 28, 2017

Abstract:  In this talk, we will review the Hopf bifurcations of planar systems. The definition of Hopf bifurcation and the Hopf bifurcation Theorem will be introduced. As an application, periodic solutions of Selkov’s model will be studied.

 

 

Talk:  Representation homology of spaces

Speaker:  Yuri Berest, Cornell University

Date: 4:00-5:00PM, Thursday, November 16, 2017

Abstract:  Let $ G $ be an affine algebraic group defined over a field $ k $. For any (discrete) group $ \pi $, the set of all representations of $ \pi $ in $ G $ has a natural structure of an algebraic variety (more precisely, affine k-scheme) called the representation variety $ Rep_G(\pi) $. If $ X $ is a (based) topological space, the  representation variety of its fundamental group $ Rep_G[π_1(X)] $ is an important geometric invariant of $X$ that plays a role in many areas of mathematics. In this talk, I will present a natural homological extension of this construction, called representation homology, that takes into account a higher homotopy information on $ X $ and has good functorial properties. The representation homology turns out to be computable (in terms of known invariants) in a number of interesting cases (simply-connected spaces, Riemann surfaces, link complements, lens spaces, ...), some of which I will examine in detail. Time permitting, I will also explain the relation of representation homology to other homology theories associated with spaces, such as higher Hochschild homology, $ S^1$-equivariant homology of  free loop spaces and the (stable) homology of automorphism groups of the free groups $ F_n $.

 

 

Talk:  Lie Symmetries of Differential Equations (II)

Speaker:  Dr. Jeongoo Cheh, The University of Toledo

Date: 4:00-5:00PM, Tuesday, November 7, 2017

Abstract:  It is well known to most students that differential equations are usually studied with tools provided by some kind of analysis -- real analysis, complex analysis, functional analysis, harmonic analysis, etc.. A very different approach is to treat differential equations as submanifolds of jet bundles and employ geometric tools to study their symmetries. In fact, it was this geometric approach to differential equations that led historically to the genesis of the vast central industry of Lie groups and Lie algebras. In this introductory talk into the area, we will start by recalling a few necessary basics on manifolds and group actions, proceed to define Lie (point) symmetries of differential equations, construct symmetry algebras and symmetry groups, and then conclude with specific examples including an application to the Hopf-Cole transformation. 

 

 

Talk:  Lie Symmetries of Differential Equations (I)

Speaker:  Dr. Jeongoo Cheh, The University of Toledo

Date: 4:00-5:00PM, Tuesday, October 31, 2017

Abstract:  It is well known to most students that differential equations are usually studied with tools provided by some kind of analysis -- real analysis, complex analysis, functional analysis, harmonic analysis, etc.. A very different approach is to treat differential equations as submanifolds of jet bundles and employ geometric tools to study their symmetries. In fact, it was this geometric approach to differential equations that led historically to the genesis of the vast central industry of Lie groups and Lie algebras. In this introductory talk into the area, we will start by recalling a few necessary basics on manifolds and group actions, proceed to define Lie (point) symmetries of differential equations, construct symmetry algebras and symmetry groups, and then conclude with specific examples including an application to the Hopf-Cole transformation. 

 

 

Talk:  Cofinite graphs and their profinite completions

Speaker:  Dr. Amrita Acharyya, The University of Toledo

Date: 4:00-5:00PM, Tuesday, October 24, 2017

Abstract: We generalize the idea of cofinite groups due to B. Hartley. First we define cofinite spaces. Then, as a special situation, we study cofinite graphs and their uniform completions. The idea of constructing a cofinite graph starts with defining a uniform topological graph $\Gamma$,  in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over $\Gamma$. It is established that for any cofinite graph there exists a unique Profinite completion.

 

 

Talk:  Integrable structures of dispersionless systems and differential geometry

Speaker: Dr. Alexandre Odesski, Brock University, Canada

Date: 4:00-5:00PM, Tuesday, October 12, 2017

Abstact: We develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we construct Gibbons-Tsarev systems associated to moduli space of algebraic curves of arbitrary genus and prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.

 

 

Talk:  Recovery of initial conditions for some classes of PDEs using discrete time samplings (II)

Speaker: Dr. Alessandro Arsie, The University of Toledo

Date: 4:00-5:00PM, Tuesday, October 3, 2017

Abstract: I will present some results about using discrete time samplings to recover in an optimal way and in suitable functional spaces the initial conditions for some classes of linear evolutive PDEs, using discrete time samplings at a fixed location. We will also provide some insights about a question posed by DeVore (Texas A&M) and Zuazua (Basque Foundation for Science) about the dependence of the optimal sampling strategy on the details of the spectrum of a linear operator. It turns out that for the class of PDEs we analyzed, the dependence of the optimal strategy on the spectrum is really weak. If time allows, I will talk about some open problems involving nonlinear PDEs (both in the integrable and non-integrable cases) and linear non-autonomous evolutionary PDEs. This is a joint paper with Roza Aceska (Ball State University) and Ramesh Karki (Indiana University East).

 

 

Talk:  Recovery of initial conditions for some classes of PDEs using discrete time samplings (I)

Speaker: Dr. Alessandro Arsie, The University of Toledo

Date: 4:00-5:00PM, Tuesday, September 26, 2017

Abstract: I will present some results about using discrete time samplings to recover in an optimal way and in suitable functional spaces the initial conditions for some classes of linear evolutive PDEs, using discrete time samplings at a fixed location. We will also provide some insights about a question posed by DeVore (Texas A&M) and Zuazua (Basque Foundation for Science) about the dependence of the optimal sampling strategy on the details of the spectrum of a linear operator. It turns out that for the class of PDEs we analyzed, the dependence of the optimal strategy on the spectrum is really weak. If time allows, I will talk about some open problems involving nonlinear PDEs (both in the integrable and non-integrable cases) and linear non-autonomous evolutionary PDEs. This is a joint paper with Roza Aceska (Ball State University) and Ramesh Karki (Indiana University East).

 

 

Talk:  A Reducibility Theorem for Smooth Quasi periodic Linear Systems

Speaker: Paduma Eranga, The University of Toledo

Date: 4:00-5:00PM, Tuesday, September 5, 2017

Abstract: In this talk, I'll explain an iterative procedure for finding a change of variables to reduce a quasi-periodic linear system into an autonomous system worked done by G.C. O'Brien. This process called the accelerated convergence method. A quasi periodic linear system is a linear system of ordinary differential equations

\begin{align*} x' & = Ax + P(\varphi)x \\ \varphi' & = \omega , \end{align*}

where $x \in \mathbb{R}^n, \varphi \in \mathbb{R}^m, \, A $  is a constant $n\times n$ matrix, $\omega $ is a constant vector in  $\mathbb{R}^m. $ P(\varphi)$ is periodic in $\varphi_i$ with period $2\pi$ for $i =1, \dots, m$.

In this discussion, we are going to obtain a quasi-periodic transformation which transform above system into the system with constant coefficients.

 

 

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Spring Semester, 2017

 

Talk:  Analysis of a Pseudo-Harmonic Tubular Bell

Speaker: Dr. Douglas Oliver, The University of Toledo

Date: 4:00-5:00PM, Tuesday April 18, 2017

Abstract: Tubular bells, or chimes are used for ambient sounds as well as serious music. Unlike most wind or stringed instruments, a tubular bell does not have a harmonic set of overtones. The lack of harmonious overtones creates a problem with using tubular bells for serious music: there is not unanimity regarding the pitch, or musical note associated with a particular tubular bell.

 

The Euler-Bernoulli model for vibrating thin beams was used to derive a mathematical model for vibrations of a tubular bell. Using this model, an analysis of the natural frequencies of a modified tubular bell was presented. One or more ends of the tubular bell were weighted with a mass. This mass changes the boundary conditions, and hence the ratio of the natural frequencies of the tubular bell.

 

Values for the ratio of the mass of weight(s) to the mass of the tube were identified such that the ratio of the frequency of the first overtone to the second overtone was 2. Under these conditions, the these overtones are one octave apart. The frequency ratios predicted by the model have been compared with experimental results of a frequency analysis of the sound produced by two physical tubes. The experimental results were in good agreement with the theoretical predictions.



Talk:  The Lavrentiev Phenomenon

Speaker: Dr. Dean A. Carlson, Mathematical Reviews, American Mathematical Society, Ann Arbor, MI

Date: 4:00-5:00PM, Tuesday, April 4, 2017

Abstract: In 1926 M. Lavrentiev gave an example of a free problem in the calculus of variations for which the infimum over the class of functions in $W^{1,1}[t_1,t_2]$ satisfying prescribed end point conditions was strictly less than the infimum over the dense subset $W^{1,\infty}[t_1,t_2]$ of admissible functions in $W^{1,1}[t_1,t_2]$. This property is now referred to as the Lavrentiev phenomenon. After Lavrentiev's discovery L.~Tonelli and B. Mania gave sufficient conditions under which this phenomenon does not arise.  After these results, the study of the Lavrentiev phenomenon lay dormant until the 1980s when a series of papers by Ball and Mizel and by Clarke and Vinter gave a number of new examples for which the Lavrentiev phenomenon occurred. Also in 1979, T. S. Angell showed that the Lavrentiev phenomenon did not occur if the integrands satisfy a certain analytic property known as property (D). Moreover, he showed that the conditions of Tonelli and Mania insured that the analytic property (D) was satisfied.  In this talk we will begin by presenting B.~Mania's elementary example to illustrate that the phenomenon exists and discuss Angell's property (D) to give a general theorem that avoids Lavrentiev's phenomenon and show briefly that some more recent results can be viewed as corollaries to Angell's result in that the conditions assumed imply property (D).

 

 

Talk:  Some classes of nonlinear integral operators and existence results via Schauder's fixed point theorem

Speaker: Dr. Alessandro Arsie, The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 28, 2017

Abstract: I will discuss three examples of nonlinear integral operators that are completely continuous on some spaces of continuous functions (they are Volterra integral operators, Fredholm integral operators and integral operators with delay). By means of Schauder's fixed point theorem, I will discuss existence of continuous solutions for the integral equations associated to these operators.

 

 

Talk:  Mathematical Modelling for Parametric Resonance

Speaker: Dr. Zhiwei Chen, The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 21, 2017

Abstract: When a physical parameter in an oscillatory system is modulated to vary in time, it may cause a dynamic instability associated with the system. This phenomenon is referred to as parametric resonance. The mathematical models amenable to such phenomena are differential equations with periodic coefficients, specifically, the Mathieu’s equation. In this talk, I will discuss some parametrically excited systems and their characteristics in resonance. I will derive some simple schemes of electrical circuits into the Mathieu’s equation and discuss the relevant analysis towards this phenomena.

 

 

Talk:  Circumference over diameter; the different universes of pi (𝝅 Day Colloquium)

Speaker: Dr. Nate Iverson, The University of Toledo

Date: 4:00-5:00PM, Tuesday, March 14, 2017

Abstract: Pi is the ratio of circumference to diameter in a circle. We define a circle to be a set of points equidistant from a common point. When the method of measuring distance is changed different ratios are possible. This talk will discuss the ratio of circumference to diameter in all p-norms including p=1, the taxicab norm, and p=∞, infinity the supremum norm. Results dating to 1932 using the Minkowski functional norms will also be discussed along with further generalizations.

 

 

Talk:  Predator-prey models with Holling types of functional responses (II)

Speaker: Dr. Chunhua Shan, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 15, 2017

Abstract: Predator-prey system has been extensively studied by biologists and mathematicians. In this talk I will introduce the classical predator-prey models of Holling types of functional responses. Dynamics of predator-prey system with Holling type II functional response will be reviewed by qualitative analysis and bifurcation theory.

 

 

Talk:  Predator-prey models with Holling types of functional responses (I)

Speaker: Dr. Chunhua Shan, The University of Toledo

Date: 4:00-5:00PM, Tuesday, February 7, 2017

Abstract: Predator-prey system has been extensively studied by biologists and mathematicians. In this talk I will introduce the classical predator-prey models of Holling types of functional responses. Dynamics of predator-prey system with Holling type II functional response will be reviewed by qualitative analysis and bifurcation theory.

 

 

Talk:  Floquet Theory and periodic linear differential equations

Speaker: Paduma Eranga, The University of Toledo

Date: 4:00-5:00PM, Tuesday, January 31, 2017

Abstract: In this talk I'll discuss a main theorem in Floquet Theory, which appear in the study of periodic linear differential equations, of the form $x' = A(t)x , A(t+T)= A(t), T>0 $ where $A(t)$ is a matrix of complex continuous functions. That main theorem; Floquet theorem due to Gaston Floquet(1883) gives a representation of a fundamental matrix solution $\Phi(t)$, as the product of periodic nonsingular matrix $P(t)$ with the same period $T$ and a constant matrix $R$ such that $\Phi(t) = P(t)e^{tR}$. As a result we can transform the periodic system into a usual linear system with constant coefficients.

 

 

Talk:  A proof of uniformly boundedness principle

Speaker: Dr. Alessandro Arsie, The University of Toledo

Date: 4:00-5:00PM, Tuesday, January 24, 2017

Abstract:  In this talk I'll discuss a main theorem in Floquet Theory, which appear in the study of periodic linear differential equations, of the form $x' = A(t)x , A(t+T)= A(t), T>0 $ where $A(t)$ is a matrix of complex continuous functions. That main theorem; Floquet theorem due to Gaston Floquet(1883) gives a representation of a fundamental matrix solution $\Phi(t)$, as the product of periodic nonsingular matrix $P(t)$ with the same period $T$ and a constant matrix $R$ such that $\Phi(t) = P(t)e^{tR}$. As a result we can transform the periodic system into a usual linear system with constant coefficients.