**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.

------------------------------------------------------------------------------------------------------------------------

**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.