# MathCS Seminar

This is the homepage of the Chapman University Mathematics and Computational Science seminar

*Seminar Organizers:* Mihaela Vajiac and Adrian Nistor

## Fall 2015

The seminar talks are in Von Neumann Hall VN 116 (545 W Palm Ave corner of W Palm Ave and railroad, Orange, CA 92866).

See [http://www.chapman.edu/discover/maps-directions/index.aspx Maps and directions], Von Neumann Hall is Building 38 on the [http://www.chapman.edu/discover/_files/CU_CampusMap2012-13-2.pdf Campus map]

### Wednesday, September 30th, 2015, at 3-5pm

#### *Speaker:* ** Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Lecture Series by Professor Daniel Alpay, Lectures 5 and 6**

*Abstract:* Rational functions are quotient of polynomials, or
meromorphic functions on the Riemann sphere. Here we consider
matrix-valued rational functions. A number of new aspects (a point can
be at the same time a zero and a pole of the function) and new notions
and methods appear (in particular the state space method. A key role
is played by the realization of a matrix-valued rational function $M$,
say analytic at the origin, that is, its representation in the form
$M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate
sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 5, 3:00pm-4:00pm

1) Wavelet filters.

2) Convex invertible cones.

3) A new kind of realization.

Lecture 6, 4:00pm-5:00pm

1) Several complex variables.

2) The non commutative case.

3) Rational functions on a compact Riemann surface, theta functions.

4) Quaternionic setting.

### Monday, September 28th 2015 at 4:00pm (tea and cookies at 3:30pm)

#### *Speaker:* ** Prof. Ahmed Sebbar, Bordeaux University, France**

*Title:* **Capacities and Jacobi Matrices**

*Abstract:* Given a system of intervals of the real line, we
construct a Jacobi matrix (tridiagonal and periodic) whose spectrum is
this given system of intervals. We discuss the underlying conditions
and techniques, as well as possible applications.

### Tuesday, September 28th, 2015, at 3:30-5:30pm

#### *Speaker:* ** Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Lecture Series by Professor Daniel Alpay, Lectures 3 and 4**

*Abstract:* Rational functions are quotient of polynomials, or
meromorphic functions on the Riemann sphere. Here we consider
matrix-valued rational functions. A number of new aspects (a point can
be at the same time a zero and a pole of the function) and new notions
and methods appear (in particular the state space method. A key role
is played by the realization of a matrix-valued rational function $M$,
say analytic at the origin, that is, its representation in the form
$M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate
sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 3, 3:30pm-4:30pm

1) Realization and geometry: $J$-unitary rational functions.

2) Applications to interpolation problems.

3) Inverse scattering problem (Krein and Marchenko).

Lecture 4, 4:30pm-5:30pm

1) First order degree systems.

2) Smith-McMillan local form.

3) Zero-pole structure.

4) Applications to inverse problems.

### Friday, September 25th, 2015, at 1-3pm

#### *Speaker:* ** Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Lecture Series by Professor Daniel Alpay, Lectures 1 and 2**

*Abstract:* Rational functions are quotient of polynomials, or
meromorphic functions on the Riemann sphere. Here we consider
matrix-valued rational functions. A number of new aspects (a point can
be at the same time a zero and a pole of the function) and new notions
and methods appear (in particular the state space method. A key role
is played by the realization of a matrix-valued rational function $M$,
say analytic at the origin, that is, its representation in the form
$M(z)=D+zC(I-zA)^{-1}B$, where $A,B,C,D$ are matrices of appropriate
sizes.

We will discuss matrix-valued rational functions, and their connections with topics such as complex analysis, interpolation theory of analytic functions contractive in the open unit disk (Schur functions), the theory of linear systems (signal processing) and matrix theory.

Lecture 1, 1pm-2pm:

1)Preliminaries on rational functions. Notion of realization.

2) Transfer functions. Link with linear systems.

3) Resolvent operators.

4) Proof of the realization theorem: { The backward-shift realization}.

5) Various characterizations of rational functions.

6) The Wiener algebra.

Lecture 2, 2pm-3pm:

1) Main properties of the realization.

2) Another proof of the realization theorem.

3) Minimal realization.

4) Minimal factorizations.

5) Spectral factorizations.

6) Reproducing kernel spaces.

### Thursday, September 17th, 2015 at 3pm (tea and cookies at 2:30pm)

#### *Speaker:* **Prof. Ahmed Sebbar, Bordeaux University, France**

*Title:* ** The Frobenius determinant theorem and applications.**

*Abstract:* In this first talk, we will discuss the celebrated determinant
Frobenius theorem and how it arised naturally in the study of a
hierarchy of hypersurfaces, of partial differential operators and
metrics.

The first elements of this hierarchy are the cubic $x^3 + y ^3 + z^3 - 3xyz = 1$ (so called Jonas hexenhut) and the partial differential operator $\Delta_3 = \frac{\delta^3}{\delta_{x^3}} + \frac{\delta^3}{\delta_{y^3}} + \frac{\delta^3}{\delta_{z^3}} -3 \frac{\delta^3}{\delta_x \delta_y \delta_z}$, introduced by P.Humbert in 1929 in another context. We explain why this operator is a good extension to ${\rm I\!R}^3$ of the Laplacian in two dimensions $\Delta_2 = \frac{\delta^2}{\delta_{x^2}} + \frac{\delta^2}{\delta_{y^2}}$ We discuss its links with Spectral theory, Elliptic functions, number theory and a sort of Finsler geometry.

This is a part of a large project conducted in collaboration with Daniele Struppa, Adrian Vajiac and Mihaela Vajiac.

### Thursday, September 3rd, 2015 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Yasushi Kondo, Kinki University, Osaka, Japan**

*Title:* **Composite Quantum Gates with Aharanov–Anandan phases.**

*Abstract:* Unitary operations acting on a quantum system must be
robust against systematic errors in control parameters for reliable
quantum computing. Composite pulse technique in nuclear magnetic
resonance realizes such a robust operation by employing a sequence of
possibly poor-quality pulses. We show that composite pulses that
compensate for a pulse length error in a one-qubit system have a
vanishing dynamical phase and thereby can be seen as geometric quantum
gates with Aharanov-Anandan phases.

## Spring 2015

### Friday, February 13th, 10:00 a.m. to noon

#### *Speaker:* **Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Fock spaces and non commutative stochastic distributions. The free setting. Free (non commutative) stochastic processes.**

*Abstract:* We present the non commutative counterpart of the
previous talk. We will review the main definitions of free analysis
required and then present, and build stationary increments non
commutative processes. The values of their derivatives are now
continuous operators from the space of non commutative stochastic test
functions into the space of non commutative stochastic distributions.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/

### Thursday, February 12th, 11:00 a.m. to 1:00 p.m.

#### *Speaker:* **Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Bochner and Bochner-Minlos theorem. Hida’s white noise space and Kondratiev’s spaces of stochastic distributions, Stationary increments stochastic processes. Linear stochastic systems.**

*Abstract:* We discuss the Bochner-Minlos theorem and build Hida’s
white noise space. We build stochastic processes in this space with
derivative in the Kondratiev space of stochastic distributions. This
space is an algebra with the Wick product, and its structure of
tallows to define stochastic integrals.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/

### Tuesday, February 10th, 10:00 a.m. to noon

#### *Speaker:* **Professor Daniel Alpay, Earl Katz Chair in Algebraic System Theory, Department of Mathematics, at Ben-Gurion University of the Negev**

*Title:* **Positive definite functions, Countably normed spaces, their duals and Gelfand triples**

*Abstract:* We survey the notion of positive definite functions and
of the associated reproducing kernel Hilbert spaces. Examples are
given relevant to the sequel of the talks. We also define nuclear
spaces and Gelfand triples, and give as examples Schwartz functions
and tempered distributions.

More details at:

http://blogs.chapman.edu/scst/2015/02/02/daniel-alpay/

### Wednesday, January 14th, 2015 at 3pm (tea and cookies at 2:30pm)

#### *Speaker:* **Prof. Richard N. Ball, University of Denver**

*Title:* **Pointfree Pointwise Suprema in Unital Archimedean L-Groups (joint work with Anthony W. Hager, Wesleyan University, and Joanne Walters-Wayland, Chapman University)**

*Abstract:* When considering the suprema of real-valued functions,
it is often important to know whether this supremum coincides with the
function obtained by taking the supremum of the real values at each
point. It is therefore ironic, if not surprising, that the fundamental
importance of pointwise suprema emerges only when the ideas are placed
in the pointfree context.

For in that context, namely in $\mathcal{R}L$, the archimedean $\ell$-group of continuous real valued functions on a locale $L$, the concept of pointfree supremum admits a direct and intuitive formulation which makes no mention of points. The surprise is that pointwise suprema can be characterized purely algebraically, without reference to a representation in some $\mathcal{R}L$. For the pointwise suprema are precisely those which are context-free, in the sense of being preserved by every $W$-morphism out of $G$.

(The algebraic setting is the category $W$ of archimedean lattice-ordered groups (`$\ell$-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every $W$-object can be canonically represented as a subobject of some $\mathcal{R}L$.)

Completeness properties of $\mathcal{C}X$ with respect to (various types of) bounded suprema are equivalent to (various types of) disconnectivity properties of $X$. These are the classical Nakano-Stone theorems, and their pointfree analogs for $\mathcal{R}L$ are the work of Banaschewski and Hong. We show that every bounded (countable) subset of $\mathcal{R}^+L$ has a join in $\mathcal{R}L$ iff $L$ is boolean (a $P$-frame). More is true: every existing bounded (countable) join of an arbitrary $W$-object $G$ is pointwise iff the Madden frame $\mathcal{M}G$ is boolean (a $P$-frame).

Perhaps the most important attribute of pointwise suprema is that density with respect to pointwise convergence detects epicity. We elaborate. Of central importance to the theory of $W$ is its smallest full monoreflective subcategory $\beta{}W$, comprised of the objects having no proper epic extensions. That means each $W$-object $G$ has a largest epic extension $G \to \beta G$, and this extension is functorial. It turns out that a $W$-extension $A \leq B$ is epic iff $A$ is pointwise dense in $B$. Thus the epireflective hull $\beta G$ of an arbitrary $W$-object $G$ can be constructed by means of pointwise Cauchy filters.

## Fall 2014

### Thursday, December 18th, 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Lander Cnudde, University of Ghent, Belgium**

*Title:* **Fourier transforms in commutative and non-commutative multicomplex settings**

*Abstract:* This seminar addresses the generalization of the
classical Fourier transform to multicomplex settings. Inspired by a
successful case study on the slices of the non-commutative Clifford
algebra $Cl_{m+1}$, a more conceptual approach to the matter is
established. Using operator relations, we construct a general
background that allows to create Fourier analogues in more general
non-commutative as well as commutative settings. Finally we illustrate
this claim and the underlying line of thoughts by setting up a Fourier
transform for the bicomplex numbers which turns out to be in
accordance to our expectations. The framework uses concepts of both
analysis and algebra, with key roles for the Mehler formula and the
Hille-Hardy formula.

### Wednesday December 10th 2014 at 4PM (tea and cookies at 3.30PM)

#### *Speaker:* **Luke Smith, Graduate Student, Department of Mathematics, University of California, Irvine**

*Title:* **Polytope Bounds on Multivariate Value Sets**

*Abstract:* Over finite fields, if the image of a polynomial map is
not the entire field, then its cardinality can be bounded above by a
significantly smaller value. Earlier results bound the cardinality of
the value set using the degree of the polynomial. However, these
bounds can be improved significantly if our bounds depend on the
powers of all monomials in a polynomial map, rather than just the one
with the highest degree. The Newton polytope of a polynomial map is
one such object constructed by each of these monomials, and its
geometry provides sharp upper bounds on the cardinality of the value
set. In this talk, we will explore the geometric properties of the
Newton polytope and show how allows for an improvement on the upper
bounds of the multivariate value set.

*Bio:* Luke Smith is a 6th year PhD student at UCI. His research
interests involves number theory, finite fields, value sets, and Witt
vectors. He also enjoys teaching and has recently been involved in
mathematics educational outreach with the UCI Math circle and MIND
Research Institute.

### Friday October 24th 2014 at 12.30 (tea and cookies at noon)

#### *Speaker:* **Dr. Brendan Fahy, Postdoctoral Fellow, KEK High Energy Research Organization, Tsukuba, JapanTBA**

*Title:* **Linear combination interpolation, Cuntz relations and infinite products (joint work with I. Lewkowicz, P. Jorgensen and D. Volok)**

*Abstract:* Calculating observable quantities in QCD at low energies
requires a non-perturbative approach. Lattice QCD is a
non-perturbative solution which quantities can be estimates using
Monte Carlo methods. However many quantities such as multi-hadron
operators require large amounts of computational power to
compute. Using the stochastic LapH method the costly matrix inverses
required are estimated rather than computed exactly drastically
reducing the computational costs. These modern computation techniques
allow for the computation of a large number of operators including
multi-hadron operators. Results of the spectrum of energies for the
lowest 50 bound states in a finite box are presented for the rho-meson
channel.

### Tuesday October 21st 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Daniel Alpay, Ben-Gurion University of the Negev, Israel**

*Title:* **Linear combination interpolation, Cuntz relations and infinite products (joint work with I. Lewkowicz, P. Jorgensen and D. Volok)**

*Abstract:* We introduce the following linear combination
interpolation problem: Given $N$ distinct numbers $w_1,..., w_N$ and $N+1$
complex numbers $a_1,..., a_N $and $c$, find all functions f(z) analytic
in a simply connected set (depending on f) containing the points
$w_1,...,w_N$ such that $\sum_{u=1}^N a_u f(w_u)=c$. To this end we prove
a representation theorem for such functions f in terms of an
associated polynomial p(z). We first introduce the following two
operations, substitution of p, and multiplication by monomials $z^j$ ,
$0<= j < N$. Then let M be the module generated by these two operations,
acting on functions analytic near 0. We prove that every function f,
analytic in a neighborhood of the roots of p , is in M. In fact, this
representation of f is unique. To solve the above interpolation
problem, we employ an adapted systems theoretic realization, as well
as an associated representation of the Cuntz relations (from
multi-variable operator theory.) We study these operations in
reproducing kernel Hilbert space): We give necessary and sufficient
condition for existence of realizations of these representation of the
Cuntz relations by operators in certain reproducing kernel Hilbert
spaces, and offer infinite product factorizations of the corresponding
kernels.

### CECHA Workshop on Integral transforms, boundary values and generalized functions, Fall 2014

Schedule: October 17th - October 21st 2014

#### Friday October 17th 2014

Chairperson: Irene Sabadini

** 10:50am-11:05am Registration/ Welcome **

** 11:05am-11:55am Michael Shapiro, Instituto Politecnico Nacional, Mexico **

Title: “On the Hilbert and Schwarz Formulas and Operators”

** 11:55am-12:10pm Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-2:50pm Mircea Martin, Baker University **

Title: “Spin Operator Theory”

** 2:50pm-3:10pm Discussion Session **

** 3:10pm-4:00pm M. Elena Luna Elizarraras, Instituto Politecnico Nacional, Mexico **

Title: “A Bicomplex Model of Lobachevsky Geometry”

** 4:00pm-4:20pm Discussion Session **

#### Saturday October 18th 2014

Chairperson: Paula Cerejeiras

** 10:00am-10:50am Matvei Libine, Indiana University Bloomington **

Title: "Geometric Properties of Conformal Transformations on $R^{p,q}$"

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Ahmed Sebbar, Institut de Mathématiques de Bordeaux **

Title: “Motions of Critical points of Green's functions”

** 11:55pm-12:00pm Discussion Session **

** 12:00pm-1:15pm Lunch, Sandhu **

** 1:30pm-2:20pm Fabrizio Colombo, Politecnico di Milano, Italy **

Title: “The Fueter-Sce Mapping and its Inverse”

** 2:20pm-2:30pm Discussion Session **

** 2:30pm-3:20pm Adrian Vajiac, Chapman University **

Title: “Multicomplex Hyperfunctions”

** 3:20pm-3:30pm Discussion Session **

#### Sunday October 19th 2014

Chairperson: Mihaela Vajiac

** 10:00am-10:50am Irene Sabadini, Politecnico di Milano, Italy **

Title: “Monogenic Hyperfunctions in One and Several Variables”

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Uwe Kӓhler, University of Aveiro **

Title: “Crystallographic structures: how to make an effective reconstruction by the spherical X-ray transform?”

** 11:55pm-12:00pm Discussion Session **

** 12:00pm-1:15pm Lunch, Sandhu **

** 1:30pm-2:20pm Paula Cerejeiras, University of Aveiro **

Title: “Diffusive Wavelets for Nilpotent Groups”

** 2:20pm-2:30pm Discussion Session **

** 2:30pm-3:20pm Daniel Alpay, Ben-Gurion University of the Negev, Israel **

Title: “Spaces of stochastic (commutative and non commutative) distributions and applications”

** 3:20pm-3:30pm Discussion Session **

#### Monday October 20th 2014

Chairperson: M. Elena Luna Elizarraras

** 10:00am-10:50am Craig Nolder, Florida State University **

Title: “Conjugate Harmonic Components of Monogenic Functions and Symmetry”

** 10:50am-11:05am Discussion Session **

** 11:05am-11:55am Graziano Gentili, Università di Firenze **

Title: “Spherical power expansion and a Mittag-Leffler theorem for semi-regular functions”

** 11:55am-12:10am Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-2:50pm Dana Clahane, Fullerton College **

Title: “Complex, Bicomplex, and Quaternionic Gaussian Moat Problems”

** 2:50pm-3:10pm Discussion Session **

** 3:10pm-4:00pm Lander Cnudde, Universiteit Gent, Belgium **

Title: “Slice Fourier transform: definition, properties and corresponding convolutions”

** 4:00pm-4:20pm Discussion Session **

** 6:30-8:30pm Social Dinner **

#### Tuesday October 21st 2014

** 10:00am-12:10pm Discussion Session **

** 12:20pm-1:30pm Lunch, Athenaeum **

** 2:00pm-4:00pm Discussion Session **

### Thursday October 9th 2014 at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Ahmed Sebbar, Institut de Mathematiques de Bordeaux**

*Title:* **On a Remarkable Power Series**

*Abstract:* We consider the sequence, defined by

$s_{2n} = s_{n}, n \geq 1; s_{2n+1} = (-1)^{n}, n \geq 0 $

or equivalently

$s_{n} = (-1)^{b} $ if $n=2^a(1+2b); a,b \in \mathbf{N}$

We explain how it is related to paperfolding and we give a precise analysis at $x = 1$ of the power series

$f(x) = \sum s_n x^n$

### Thursday, September 25th 2014, at 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Christopher Lyon, CalState Fullerton**

*Title:* **Two notions of mirror symmetry for certain K3 surfaces**

*Abstract:* In the mid-1990s, the physicists Berglund and Hubsch
proposed a way to construct a ``mirror partner* for certain kinds of*
Calabi-Yau manifolds. When the manifold has (complex) dimension 2,
these are examples of K3 surfaces. Around the same time, Dolgachev
and others conceived of a version of mirror symmetry that applies to
more general families of K3 surfaces. In this talk, we will introduce
these special kinds of K3 surfaces, which are defined as hypersurfaces
in weighted projective space. Then we will discuss the issue of
compatibility between the aforementioned versions of mirror symmetry.
While the question is open in general, we will highlight a particular
collection of surfaces where the compatibility can be proved. This is
joint work with Paola Comparin, Nathan Priddis, and Rachel Webb.

### Thursday, September 11th 2014, 4pm (tea and cookies at 3:30pm)

#### *Speaker:* **Prof. Ahmed Sebbar, Institut de Mathematiques de Bordeaux**

*Title:* **Equivariant functions**

*Abstract:* An equivariant function is a special meromorphic
function on the Poincare upper half-plane. A concrete non trivial
example was given by Don Zagier answering a question of the physicist
Werner Nahm. We show how to construct all the equivariant functions
by using ideas from complex analysis, modular forms and projective
differential geometry. The talk is based on a joint work with
Abdellah Sebbar from The university of Ottawa.