# MathCS Seminar 2012

*Seminar Organizers:* Peter Jipsen and Cyril Rakovski

## Fall 2012

### Thursday, November 15, 2012 at 2 pm - 5 pm in VN

#### Third Chapman University and Cal State University Fullerton Joint Mathematics Colloquium

held at **Chapman University, Von Neumann Hall** (545 W. Palm Ave, Orange)
meet at 1:30 pm in Von Neumann Hall for a brief reception

**PROGRAM**

**2:00 pm Lucy Odom (CSU Fullerton)** - "Amalgamatic Curvature and Absolute Umbilical Hypersurfaces"

**2:25 pm David Tyler (Chapman University)** - "Map the Network"

**2:50 pm Charley Conley (CSU Fullerton)** - "A Rigidity Theorem for Hypersurfaces"

**3:15 pm Adrian Vajiac (Chapman University)** - "Bicomplex Numbers"

**3:40 pm Coffee Break**

**4:10 pm Nicholas Salinas (CSU Fullerton)** - "Polynomial Isometries of P-adic Integers"

**4:35 pm Peter Jipsen (Chapman University)** - "A Category of Algebraic Contexts equivalent to Idempotent Semirings"

### Monday, October 22 -- Friday, October 26 2012

#### Workshop on Function Theories for Bicomplex and Hyperbolic Numbers

Hosted by the **Center of Excellence in Complex and Hypercomplex Analysis (CECHA)**

**PROGRAM**

**Monday October 22nd 2012**

9:30am-10:30am **Adrian Vajiac, Chapman University**

Title: “Bicomplex Analysis: an introduction”

10:45am-11:45am **Michael Shapiro, Instituto Politecnico Nacional, Mexico**

Title: “Bicomplex Derivability, Differentiability, and Holomorphy”

2:00pm-3:00pm **John Ryan, University of Arkansas**

Title: “Complex Clifford Analysis”

3:15pm-4:15pm **Craig Nolder, Florida State University**

Title: “Conformal Mappings in R1,1 are Quasi-Conformal”

**Tuesday October 23rd 2012**

9:30am-10:30am **Elena Luna Elizarraras, Instituto Politecnico Nacional, Mexico**

Title: “On Fundamentals of Functional Analysis with Bicomplex Scalars”

10:45am-11:45am **Dominic Rochon, Université du Québec à Trois-Rivières, Canada**

Title: “Infinite Dimensional Bicomplex Hilbert Spaces“

2:00pm-3:00pm **Daniele Struppa, Chapman University**

Title: “Holomorphic functions of a bicomplex variable and Ehrenpreis’ Fundamental Principle”

3:15pm-4:15pm **Franciscus Sommen, Universiteit Gent, Belgium**

Title: “Spherical Monogenics on the Lie Sphere”

**Wednesday October 24th 2012**

9:30am-10:30am **Uwe Kahler, University of Aveiro, Portugal**

Title: “Discrete Function theories in the case of bicomplex and hyperbolic numbers“

10:45am-11:45am **Paula Cerejeiras, University of Aveiro, Portugal**

Title: “On the Gabor Transform: the Bicomplex Case”

**Thursday October 25th 2012**

9:30am-10:30am **Daniel Alpay, Ben-Gurion University of the Negev, Israel**

Title: “Schur Analysis and Theory of Linear Systems: The Bicomplex Case“

10:45am-11:45am **Sebastien Tremblay, Université du Québec à Trois-Rivières, Canada**

Title: “Hyperbolic Pseudonanalytic Functions in Mathematical Physics“

2:00pm-3:00pm **Irene Sabadini, Politecnico di Milano, Italy**

Title: “Algebraic Analysis Methods in the Bicomplex setting “

3:15pm-4:15pm **Fabrizio Colombo, Politecnico di Milano, Italy**

Title: “Bicomplex Holomorphic Functional Calculus “

**Friday October 26th 2012**

9:30am-10:30am **Matvei Libine, Indiana University**

Title: “Introduction to Split Quaternionic Analysis “

10:45am-11:35am **Mihaela Vajiac, Chapman University**

Title: “Hyperbolic Analysis: an introduction”

### Thursday, October 4, 2012 at 1:15 pm in VN 116

*Speaker:* **Prof. Andrew Moshier, Faculty of Computer Science and Mathematics, Chapman University**

*Title:* **Generalizing Modal Logic for Compact Hausdorff Semantics**

*Abstract:* Normal modal logic can be regarded as a specification language for non-deterministic
state transition systems, modeled as compact, zero-dimensional topological spaces equipped
with a closed binary relation. The zero-dimensionality requirement, however, severely limits
potential applications. After all, this rules out such state spaces as spheres, tori and other
garden variety compact Hausdorff spaces.

In this talk we consider how to generalize descriptive frames to allow for compact Hausdorff state spaces, and then ask how modal logic can be adapted to this generalization. In the result, we show that a Sahlqvist Theorem holds, characterizing a class of formulas that determine first-order definable properties of the state transition.

This is joint work with Nick Bezhanishvili.

### Tuesday, August 28, 2012 at 4 pm in VN 116

*Speaker:* **Dr. Petr Vojtechovsky, Department of Mathematics, University of Denver**

*Title:* **Computational aspects of loop theory**

*Abstract:* Loop theory is closely related to group theory, but the lack of associativity makes calculations with loops difficult. Since 2000, there has been notable progress in several areas of loop theory thanks to a variety of computational tools, including group-theoretical packages, symbolic computation systems, automated theorem provers and finite model builders. In this talk I will present four projects where these computational tools were essential, although one would probably not have guessed it from the results alone. I will not assume any prior knowledge of loop theory in the talk.

## Spring 2012

### Thursday, May 10, 2012 at 4 pm in VN 116

*Speaker:* **Dr. Alexander Kurz, University of Leicester, UK**

*Title:* **Categories definable by operations and equations**

*Abstract:* It is widely accepted that the category theoretic notion of a monad captures abstractly the idea of an algebraic theory. Indeed, a theorem of Linton 1969 says that the algebras for any monad can be represented by operations and equations if we are willing to generalise the notion of arity. In the talk, we are going to recall the notion of monad and the essence of Linton's argument before giving an application to coalgebras and modal logic (which is joint work with J. Rosicky).

### Friday, April 20, 2012, 2 pm CSU Fullerton

**Second Chapman University and Cal State University Fullerton joint Mathematics Colloquium** held at **CSU Fullerton, McCarthy Hall 476**
meet at 1:30 pm in McCarthy Hall 484 for a brief reception

**PROGRAM**

**2:00 pm John Simanyi (CSUF undergraduate student) - Complex Series at the Radius of Convergence**

*Abstract:* In complex analysis, as in the real case, we can find a radius of convergence for a given series, dividing the complex plane into separate regions of certain convergence and divergence. What happens at the border, on the circle of precisely that radius? We consider a few examples to investigate.

**2:30 pm Dr. Mark Filowitz (Associate Dean) - Welcoming Address**

**2:35 pm Alex Barrett (Chapman University, undergraduate student) - A Two-Light Version of the Classical Hundred Prisoners and a Light Bulb Problem: Optimizing Experimental Design through Simulations**

*Abstract:* We propose five original strategies of successively increasing complexity and efficiency that address a novel version of a classical mathematical problem that, in essence, focuses on the determination of an optimal protocol for exchanging limited amounts of information among a group of subjects with various prerogatives. The inherent intricacy of the problem–solving protocols eliminates the possibility to attain an analytical solution. Therefore, we implemented a large-scale simulation study to exhaustively search through an extensive list of competing algorithms associated with the above-mentioned 5 generally defined protocols. Our results show that the consecutive improvements in the average amount of time necessary for the strategy-specific problem-solving completion over the previous simpler and less advantageously structured designs were 18, 30, 12, and 9% respectively. The optimal multi-stage information exchange strategy allows for a successful execution of the task of interest in 1722 days (4.7 years) on average with standard deviation of 385 days. The execution of this protocol took as few as 1004 and as many as 4965 with median of 1616 days.

**3:00 pm Heng Sok (Chapman University, undergraduate student) - On the Performance of Exact Testing Procedures with Respect to Comparisons of Several Multinomial Distributions in Small Samples**

*Abstract:* This research project focuses on the validity of the exact p-value method under the fully specified and non-specified null hypotheses with respect to comparison of multiple multinomial distributions in small samples, where classical chi-square testing procedure is inappropriate. Here by validity, we mean the type I error rates at several significance levels and the power of the test under various alternatives. The two sample comparison problem arises often in two treatment randomized trials and case-control studies where comparison of background characteristics is an important step of the study analysis. Small samples are an inherent trait of pilot studies and studies with recruitment difficulties due to rare conditions, lack of interest and consent or budget and time restrictions. Further, this analysis could be extended to encompass the comparison of more than two multinomial distributions that would be applicable to multiple treatment randomized trials and as done in the classical large sample theory. Our results show that under the non-specified null hypothesis, the exact p-value method is severely conservative due to the absence of adjustment structure in these tests due the estimation of the common parameters and predictably, as the number of sample increases, the type 1 error rate increases but still remains below its nominal level. Consequently, the power of the test is low when the underlying multinomial probability distributions are relatively close to each other and the number of categories is large compared to the sample size.

**3:30 pm Nicholas Blackford, Daniel Lenders, and Danny Orton - An Inverse-Based Analogue of the Probability that Two Elements of a Finite Group Commute**

*Abstract:* Given a finite group G, the probability that two randomly chosen elements of G commute has long been viewed in the literature as a natural measure of the degree of commutativity enjoyed by the group. Many variants on this probabilistic question have arisen in the literature recently, and our research introduces yet another such variation that provides somewhat different information. In particular, given a product of elements of G, we investigate the likelihood of being able to permute the order of the elements in the product and obtain the inverse of the original element. With the help of the software program Groups, Algorithms and Programming (GAP) , we have discovered patterns leading to interesting results about this variant. In our paper, we will examine these patterns for such familiar finite groups as cyclic, dihedral, and symmetric groups, and describe the general results we have obtained. Some of these results are at odds with analogous ones known for the commutativity measure studied in recent articles, thereby adding further interest into our investigation.

### Friday, April 20, 2012, 1:30 pm in VN 116

*Speaker:* **Dr. Lucas Layton**,

*Title:* **Where the rubber meets the road: How software engineering research can benefit computational science**

*Abstract:* Computational science and software engineering are rarely spoken in the same breath. Heavyweight software development processes are not suited to the dynamic nature of scientific exploration. In this talk, I will present how research on modern software engineering tools, developer practices, and software measurement can be applied to the computational sciences to increase productivity and enable lightweight management of development efforts. I will also discuss the unique research opportunities the Chapman environment can provide to students in software engineering, and how Chapman provides a unique setting studying the important yet neglected intersection of computational science and software engineering.

### Tuesday, April 17, 2012 at 4pm in VN 116

*Speaker:* **Dr. Erik Linstead, School of Computational Sciences, Schmid College of Science, Chapman University**

*Title:* **Latent Topic Models for Automated Program Comprehension**

*Abstract:* Software development is a complex and multi-faceted process. In recent years, the open-source movement has made available for the first time an abundance of dynamic software trajectories via the public archiving of software releases. This has fueled new directions of research aimed at visualizing and better understanding software and the people who create it.

Automated topic modeling has been successfully used in text mining and information retrieval where it has been applied to the problem of summarizing large text corpora. Recent techniques include Latent Dirichlet Allocation (LDA), which probabilistically models text documents as mixtures of latent topics, where topics correspond to key concepts presented in the corpus. These more recent approaches have been found to produce better results than traditional methods such as latent semantic analysis. Yet despite the capabilities of statistical topic models, applications have typically been limited to traditional text corpora such as academic publications, news reports, and historical documents. At the most basic level, however, a code repository can be viewed as a text corpus, where source files are analogous to documents and tokens to words. Though vocabulary, syntax, and conventions differentiate a programming language from a natural language, the tokens present in a source file are still indicative of its concerns (ie. its topics).

In this talk we adapt and apply latent topic models to automated program understanding, bug triage, software evolution, and static analysis of software complexity. In particular we provide, for the first time, a formal definition of software scattering and tangling on which to base new software quality metrics. The result of this work is an unsupervised, probabilistic, machine learning framework capable of supporting software analysis on an Internet-scale. Empirical results are presented for several well-known and widely studied open-source software projects, and future research directions in the area of search-driven software development are discussed.

### Thursday, April 12, 2012 at 4pm in VN 116

*Speaker:* **Professor Greg Fasshauer**, **Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL**

*Title:* **Computing with Positive Definite Kernels**

*Abstract:*
Positive definite kernels and their associated reproducing kernel Hilbert spaces
provide a very flexible and powerful tool for the solution of many typical problems of
numerical analysis and scientific computing such as function approximation, numerical
integration or the numerical solution of PDEs. I will provide an introduction to positive
definite reproducing kernels, mention some applications, and end with a discussion of
some future research directions. This talk should be accessible to upper-level
undergraduate students and beginning graduate students.

### Tuesday, April 10, 2012 at 4pm in VN 116

*Speaker:* **Dr. Boyana Norris, Mathematics and Computer Science Division, Argonne National Laboratory, Chicago, IL**

*Title:* **Automatic Differentiation in Computational Science**

*Abstract:*
Derivatives play an important role in a wide variety of scientific computing applications, including numerical optimization, solution of nonlinear equations, sensitivity analysis, and nonlinear inverse problems. Automatic differentiation (AD) enables the accurate computation of derivatives via semantic transformation of source code via the application of the rules of differential calculus. By using AD, accurate derivatives can be obtained at a small multiple of the cost of computing the original function, which makes AD more efficient than manual parameter perturbation and finite-difference-based calibration. AD has been widely used in applications in the physical, chemical, biological, and social sciences.

In this talk, I will introduce AD and describe our approach to enabling efficient automatic differentiation of computational science applications. Our source-transformation AD tools are implemented by using a component-based software engineering approach, enabling reuse of complex capabilities, such as program analysis and the differentiation transformations, while providing separate parsing and code generation components for C, C++, and Fortran. The differentiation process consists of the following steps implemented as loosely coupled components: parsing of the arbitrarily large application source code; performing a number of program analyses; creating a standard, XML-based language-independent intermediate representation of the computational graph; augmenting the computational graph with derivative computations; converting the resulting graph back into the source language's intermediate representation; and finally, producing source code for computing the original function and its derivatives. In addition to existing capabilities, I will discuss a number of open research issues, ranging from mathematical considerations to software engineering challenges in enabling robust and efficient AD in an environment of ever-increasing software and hardware complexity.

### Thursday, March 22, 2012 at 4pm in VN 116

*Speaker:* **Dr. Mihaela Vajiac, Faculty of Mathematics, Chapman University**

*Title:* **Examples of Functional Calculus**

*Abstract:* In this talk, I will present examples of Functional Calculus, a subject introduced recently by F. Colombo, I. Sabadini, and D. Struppa, for commuting and non-commuting operators, using different classes of functions. The theory is quite new and lacks examples, a gap which this work is attempting to fill.

### Tuesday, March 20, 2012 at 4pm in VN 116

*Speaker:* **Eddy Sfeir, Deutsche Bank, and Chapman University Alumnus**

*Title:* **Trading Foreign Exchange Options**

*Abstract:* Out of all markets, the currency market is the biggest out there with over 3 Trillion USD exchanging hands every day. From the basic Black-Scholes formula to slightly more advanced concepts such as delta and gamma hedging, the purpose of this talk is to show you what a day in the life of a currency options trader consists of.

Eddy Sfeir has been trading options for Deutsche Bank for nearly 4 years. He started out on Wall St, and is now trading in London. Deutsche Bank has been number 1 in FX for 7 years in a row (in terms of market share, customer surveys etc.)

### Thursday, March 15, 2012 at 4pm in VN 116

*Speaker:* **Professor Atanas Radenski, School of Computational Sciences, Schmid College of Science, Chapman University**

*Title:* **Distributed Simulated Annealing with MapReduce on the Cloud**

*Abstract:* Simulated annealing’s high computational intensity has stimulated researchers to experiment with various annealing algorithms for shared memory, message-passing, and hybrid-parallel platforms. MapReduce is an emerging distributed computing framework for large-scale data processing on clusters of commodity servers; to our knowledge, MapReduce has not been used for simulated annealing yet. In this talk, we investigate the applicability of MapReduce to distributed simulated annealing in general, and to the TSP in particular. We (i) outline six algorithmic patterns of distributed simulated annealing with MapReduce, (ii) discuss the instantiation of such patterns into MR implementations to solve sample TSP problems, and (iii) evaluate solution quality and speedup of the implementations on a cloud computing platform, Amazon’s Elastic MapReduce. The talk can be beneficial for those interested in the potential of cloud-based MapReduce in computationally intensive methods in general and nature-inspired algorithms in particular.

### Thursday, February 23, 2012 at 12 noon in VN 116

*Speaker:* **Dr. Roman Buniy, Faculty of Physics, Chapman University**

*Title:* **New measures for entangled states**

*Abstract:* We propose new measures that distinguish and classify entangled states. The measures are algebraic invariants of linear maps associated with the states. Considering qubits as well as higher spin systems, we obtained complete entanglement classifications for cases that were either unsolved or only conjectured in the literature.

### Wednesday, February 1, 2012 at 4:00 pm in VN 116

*Speaker:* **Professor Aleš Pultr, Charles University, Prague, Czech Republic**

*Title:* **Algebraic and Topological Aspects of the Coproduct of Frames**

*Abstract:* The algebraic aspects concern the coproduct as the tensor product in a broader category.
This is the same phenomenon as the coproduct of commutative rings being
carried by the tensor product of the underlying abelian groups, and we will
consider how it is in a general setting.

The topological aspects are related to confronting the coproduct $\Omega(X)\otimes\Omega(Y)$ with the product $X\times Y$.

The consequences of the discrepancy are generally pleasant, as can be seen in the behaviour of paracompact frames, the closed subgroup theorem, and a pleasant situation in confronting cover and entourage uniformity.

### January 9 - January 19, 2012 in VN 116

**FOURTH ANNUAL CECAT WORKSHOP IN POINTFREE TOPOLOGY & ANALYSIS**

Monday 9th: | 12pm-2pm | Bernhard Banaschewski (McMaster University, Canada)
| |

"Pseudocompactness in Frames" | |||

2.30pm-3.30pm | Drew Moshier (Chapman University)
| ||

"D-frames" | |||

Tuesday 10th: | 10am-12pm | Drew Moshier
| |

"D-frames" | |||

12.30pm-2.30pm | Bernhard Banaschewski
| ||

"Pseudocompactness in Frames" | |||

Wednesday 11th: | No scheduled talks | ||

Thursday 12th: | 11am-1pm | Drew Moshier
| |

"D-frames" | |||

1.30pm-3.30pm | Bernhard Banaschewski
| ||

Friday 13th: | 10.30am-noon | Bernhard Banaschewski
| |

"The rings of continuous functions on sigma frames" | |||

12.30pm-2pm | Drew Moshier
| ||

"A relational category of formal contexts" | |||

Saturday 14th: | 10.30am-noon | Bernhard Banaschewski
| |

"The rings of continuous functions on sigma frames" | |||

12.30-2pm | Rick Ball (University of Denver, joint work with Tony Hager)
| ||

"Truncated and diminished l-groups" | |||

2.30pm-4pm | Garth Dales (University of Lancaster, UK)
| ||

"The hyper-Stonean cover of a compact space" | |||

Sunday 15th: | 10.30am-noon | Bernhard Banaschewski
| |

"The rings of continuous functions on sigma frames" | |||

12.30-2pm | Rick Ball (joint work with Tony Hager)
| ||

"Truncated and diminished l-groups" | |||

4pm | Cello Concert followed by informal reception
| ||

Monday 16th: | 10am | Coffee at Café Lucas | |

11am-1pm | Rick Ball (joint work with Tony Hager)
| ||

"Truncated and diminished l-groups" | |||

1.30pm-3pm | Peter Jipsen (Chapman University)
| ||

"On the duality between lattices and topological contexts" | |||

Tuesday 17th: | 10am-12pm | Bernhard Banaschewski
| |

"TBD" | |||

12:30pm-2pm | Peter Jipsen
| ||

"On the duality between lattices and topological contexts" | |||

Wednesday 18th: | No scheduled talks | ||

Thursday 19th: | 11am-2.30pm | Open session
| |

"D-frames" |