# MathCS Seminar 2007

## Fall 2008

The seminar talks are usually in Beckman Hall 207 (corner of N. Glassell St and University Drive, Orange, CA 92866)

### Friday, December 5th, at 1:00pm in BK207

*Speaker:* Dr.Martha Dussan, Universidade de Sao Paulo & UCI

*Title:*
A characterization of Christoffel transform in the Split-Quaternions

*Abstract:*
This talk is focusing to characterize the Christoffel pairs of timelike isothermic surfaces in the split-quaternions $H' = \mathbb R^4_2$. When we restrict the ambient space to the imaginary split-quaternions $Im H' = \mathbb R^3_2$, we also characterize that kind of pair through of the existence of an integranting factor. This a joint work with Prof. M. Magid.

### Thursday, November 20th, at 1:00pm in BK207

*Speaker:* Dr. David Pincus ,
Chapman University,

*Title:*
Fractal Dynamics in Interpersonal Processes

*Abstract:*
This presentation will review a line of research investigating a theoretical model of interpersonal dynamics based on processes of self-organization and complexity theory. Empirical investigations of the complexity of patterns of repetition in verbal turn-taking behaviors during conversations using various measures of entropy (topological, information, and fractal dimension) have consistently demonstrated that interpersonal dynamics exhibit fractal patterns characteristic of far-from-equilibrium conditions at the “edge-of-chaos.” For example, statistical fits between an inverse-power-law (IPL) model and long-sequence patterning in conversations have ranged from R2 = .87 to R2 = .99. These results have been found in conversations within group therapy, family therapy, and experimentally-created relationships among strangers. Furthermore, a statistical model of interpersonal closeness, conflict, and control accounted for a combined 48% of the variance in pattern repetition (structure) within the IPL above and beyond speaker base-rates (combinatorial probabilities). Finally, a series of experimental investigations has demonstrated that the fractal dimension of turn-taking patterns within small groups shifts significantly in the direction of rigidity depending upon levels of internal conflict within group members. In addition to some practical significance for applied psychology, these results provide for a number of possible theoretical integrations, within psychology between psychology and other scientific domains. Key areas for discussion will include: a) The potential to derive more formal and specific mathematical models (e.g., differential equation models) to simulate relationship development over time and under various conditions; b) Graphics programs that could more effectively display the fractals underlying these conversations; and c) improved statistical procedures that could better identify processes such as bifurcations, transients, and changes in the contributions of individuals to overall group structure during conversations.

### Thursday, November 6th, at 4:00pm in BK207

*Speaker:* Dr. Adrian Vajiac,
Chapman University,

*Title:*
Algebraic Analysis of ElectroMagnetism

*Abstract:*
In recent years, techniques from computational algebra have become important to render effective general results in the theory of Partial Differential Equations. Following our work with D.C. Struppa, I. Sabadini, F. Colombo, and M. Vajiac, we will present how these tools can be used to discover and identify important properties of several physical systems of interest such as Electromagnetism and Abelian Instantons.

### Thursday, October 23rd, at 4:00pm in BK207

*Speaker:* Professor Roman A. Polyak,
Department of Mathematical sciences & SEOR Department, George Mason University

*Title:*
Nonlinear Equilibrium vs. Linear Optimization

*Abstract:*
We consider the Generalized Walras-Wald Equilibrium (GE) as an alternative to Linear Programming (LP) approach for optimal recourse allocation. There are two fundamental differences between the GE and LP approach for the best resource allocation.
First, the prices for goods (products) are not fixed as they are in LP; they are functions of the production output.
Second, the factors (resources) used in the production process are not fixed either; they are functions of the prices for the resources.
It was shown that under natural economic assumptions on both prices and factors vector functions the GE exists and unique.
Finding the GE is equivalent to solving a variational inequality with a strongly monotone operator on nonnegative octants of the primal and dual spaces. For solving the variational inequality a projected pseudo-gradient method was introduced , his global convergence with Q-linear rate was proven an its computational complexity was estimated.
The method can be viewed as a natural pricing mechanism for establishing an economic equilibrium.

About this week's speaker: Roman A.Polyak, received Ph. D in mathematics from Moscow Central Institute of Mathematics and Economics at the Soviet Academy of Sciences. After emigration from the former Soviet Union he was a visiting scientist at the Mathematical Sciences Department at the T.J. Watson Research Center IBM. Since 1995 Dr. Polyak is a Professor of Mathematics and Operations Research at George Mason University. He is an author and co-author of six monographs and chapters of books and published more than sixty papers in refereed professional journals. His area of expertise is Linear and Nonlinear programming , game theory and mathematical economics. He received several NSF and NASA Awards as well as the Fulbright Scholarship Award for his work on Nonlinear Rescaling Theory and Methods in constrained optimization.

### Thursday, October 16th, at 4:00pm in BK207

*Speaker:* Dr. Mihaela Vajiac,
Chapman University,

*Title:*
Hartogs Phenomena and Antisyzygies for Systems of Differential Equations

*Abstract:*
This work, in collaboration with A. Damiano, D. Struppa, and A. Vajiac introduces the notion of antisyzygies, which studies the inverse problem of finding a system of PDEs, given compatibility conditions. The system obtained possesses the property of removability of compact singularities. We also write explicit computations in the cases of the Cauchy-Fueter system and Maxwell’s system for electromagnetism, and we conclude with a study of systems of non-maximal rank.

### Thursday, October 2nd, at 4:00pm in BK207

*Speaker:* Dr. Cyril Rakovski,
Chapman University,

*Title:*
Unweighted analysis of counter-matched case-control data.

*Abstract:*
Informative sampling based on counter-matching risk set subjects on an exposure variable has been shown to be an efficient alternative to simple random sampling when the counter-matching variable is correlated with the variable of interest; however, the opposite is true when the counter-matching variable is independent of the variable of interest. For given counter-matched data, we consider a naive analysis of the effect of a dichotomous covariate on the disease rates that ignores the underlying sampling design and its corresponding effect on the analytic expression of the partial likelihood. We provide analytical expressions for the bias and variance and show that under mild common conditions such an analysis is clearly advantageous over the standard "weighted" approach. The efficiency gains could be large and are inversely related to the prevalence of the counter-matching variable. Finally, we ascertain that departures from the required assumptions entail biased estimates and provide numerical values for the bias for common scenarios.

### Thursday, September 25th, at 4:00pm in BK207

*Speaker:* Prof. Arek Goetz,
San Francisco State University

*Title:*
The interplay of symbolic computing with geometry in dynamical systems arising from theoretical electrical engineering problems.

*Abstract:*
The talk accessible to nonprofessionals will lead the audience into an exciting journey of the beauty of geometric structures obtained using computational mathematics derived from discontinuous dynamical systems that arise from problems in electrical engineering.
We illustrate how rigorous computational mathematics and elementary number theory is used producing fractal structure of piecewise isometries.

Key words: fractals, computational mathematics, cyclotomic fields, dynamical systems, digital filters, piecewise isometries, computers producing publishable papers.

The CV link: http://calculus.sfsu.edu/goetz/vitae.pdf Arek Goetz is professor of mathematics at San Francisco State University. An active researcher in dynamical systems, software architect and an educator, has delivered over 90 talks on 5 continents. He is a recipient of two National Science Foundation Grants, as well as numerous teaching grants.

### Thursday, September 11th, at 4:00pm in BK207

*Speaker:* Giovanni Sambin
Department of Pure and Applied Logic
University of Padua

*Title:*
The Basic Picture: an interactive introduction by questions & answers.

*Abstract:*
To define a topology on a set $X$ in a constructive way (where contructive here means both intuitionistic and predicative), one needs a second set $S$ and a family of subsets of $X$ indexed on $S$, that is, a relation between $X$ and $S$. I have called this structure a basic pair.
One can show that interior and closure of a subset are defined in a basic pair by formulae which are strictly dual of each other (the former is of the form $\exists\forall$, the latter $\forall \exists $).

Continuity between basic pairs is expressed by a commutative diagram of relations (up to a suitable notion of equality).

The main pointfree structure has a primitive for closed subsets (positivity) which is dual to that for open subsets (formal cover), and they are linked by a condition (called compatibility) which is best expressed by using the notion of overlap between subsets (the existential dual of inclusion).

These discoveries show that there is a clear and simple structure underlying topology, and that it is a sort of applied logic. I have called it the Basic Picture. Both traditional ("pointwise") and pointfree topology in the proper sense are obtained as a special case. In fact, a topological space is just a basic pair equipped with convergence (any two approximations of a point can be refined to a third), and continuous functions are just those relations which preserve convergence.

Besides allowing for a fully constructive development of topology, this approach brings to some technical improvements which are new also for the classical approach. In particular, one can prove that the category of topological spaces with continuous relations can be embedded in the category of formal topologies (i.e. pointfree topologies in a constructive sense), thus giving a mathematical form to the well-known claim that pointfree topology generalizes the traditional one with points.

On all this, I am writing a book (The Basic Picture. Structures for constructive topology", Oxford U.P., forthcoming). After around 20 minutes expanding on the above summary, I am ready to give an introduction to the content and general philosophy of the book, by discussing with the audience any questions they would like to put.

### Thursday, August 28, 2008, 4-5 pm, BK 207

*Speaker:* Dr. Georg Struth (University of Sheffield, UK)

*Title:* Verification Based on Algebra and Automated Deduction

*Abstract:* The formal analysis and verification of computing systems has so far been dominated by model checkers and other decision procedures which are fully automated, but limited in expressive power, and by interactive theorem provers which are quite expressive, but limited in automation. Due to improved hardware and theoretical developments, automated deduction is currently emerging as a third way in which expressivity and computational power are differently balanced and which conveniently complements the other approaches.

I will present a new approach to formal verification in which computational algebras are combined with off-the-shelf automated theorem provers for first-order equational logic. The algebras considered are variants of Kleene algebras and their extensions by modal operators. Particular strengths of these structures are syntactic simplicity, wide applicability, concise elegant equational proofs, easy mechanizability and strong decidability.

I will sketch the axiomatization and calculus of Kleene algebras and modal Kleene algebras, discuss some computationally interesting models, such as traces, graphs, languages and relations, and point out their relationship to popular verification formalisms, including dynamic logic, temporal logic and Hoare logic. I will also report on some automation results in the areas of action system refinement, termination analysis and program verification that demonstrate the benefits and the potential of the algebraic approach.

## Spring 2008

### Thursday, May 8, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Professor Alfonso Cariazzo (University of Sevilla)

*Title:* Chen's inequalities in generalized Sasakian-space-forms

*Abstract:* Recently, P. Alegre, D. E. Blair and the speaker defined generalized Sasakian-space-forms as those almost contact metric manifolds with a Riemann curvature tensor satisfying the usual equation for a Sasakian-space-form, but with some differentiable functions $f_1; f_2; f_3$ instead of the well-known constant quantities (c + 3)=4 and (c ¡ 1)=4.
In this talk, we will review the main facts about generalized Sasakian-space-forms, such as the existence of interesting examples in any dimension, or the possible structures for these spaces. After that, we will present sharp inequalities involving ±-invariants for submanifolds in this setting, with arbitrary codimension. In fact, ±-invariants, introduced by B.-Y. Chen, have proven to be a key tool in Submanifolds Theory, providing new very useful information concerning the immersion problem.

### Thursday, May 1, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Professor Vincenzo Vespri, (University of Firenze)

*Title:* Harnack inequalities for nonlinear parabolic equations

*Abstract:* In this talk I'll speak about recent results obtained in collaboration with Emmanuele DiBenedetto and Ugo Gianazza.
In the sixties Moser, using deep Nash ideas, proved Harnack inequalities for nonnegative solutions of linear parabolic equations with $L^\infty$ coefficients. This approach, however, seems not to work in the nonlinear case (for instance in the case of p-Laplacean and porous medium equation). In recent papers published on Calc. Var., Acta Math and Duke Math. J., we give an alternative proof of the Harnack inequality with respect the Moser's one based on DeGiorgi's function classes. This approach is so flexible that it can be extended to the nonlinear case.

### Tuesday, April 29, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Hung-Jen (Steward) Huang, (Chapman University)

*Title:* Bayesian Analysis of Errors-in-Variables Growth Curves with Skewness in Models

*Abstract:* We propose to analyze model data 1) using errors-in-variables (EIV) model and 2) using the assumptions that the error random variables are subject to the influence of skewness through Bayesian approach. The use of EIV in model is necessary and realistic in studying many statistical problems, but their analysis usually mandate many simplifying and restrictive assumptions. Previous studies have shown the superiority of Bayesian approach in dealing with the complexity of these models.

In fitting statistical models for the analysis of growth data, many models have been proposed. We selected an extensive list of the most important growth curves and using some of them in our model analysis. Much research using classical approach has clustered on this area. However, the incorporation of EIV into these growth models under Bayesian formulation with skewness models have not yet been considered or studied. A motivating example is presented and in which we expose certain lacunae in the analysis previously done as well as justify, the applicability of the our general approach proposed alone. In addition, auxiliary covariates, both qualitative and quantitative, can be added into our model as an extension. This EIV growth curves with auxiliary covariates in models renders a very general framework for practical application.

Another illustrative example is also available to demonstrate how Bayesian approach through MCMC (Metropolis Hastings/slice sampling in Gibbs sampler) techniques as well as Bayesian Information Criterion (BIC) for model selection can be utilized in the analysis of this complex EIV growth curves with skewness in models.

### Thursday, April 24, 2008, 2:15-3:15 pm, BK 207

*Speaker:* Alberto Damiano (Chapman University)

*Title:* Syzygies, Antisyzygies, and how I defeated the Dirac system in a Singular way

*Abstract:* In my talk I will present an overview of some of the results from this year’s collaboration with D. Struppa, and A. and M. Vajiac, together with some of the questions that still remain open. First, I will introduce a new concept of “antisyzygies”, which constitutes a sort of inverse problem within the framework of algebraic analysis. Usually people start from a (non-homogeneous) system of PDEs, and hunt for the integrability conditions. We asked ourselves the opposite question: if we are given the compatibility laws, how can we reconstruct the system in some canonical way? What are the properties of the system that we get? In particular, I will mention the relation between the antisyzygy construction and Hartogs type of phenomena (removability of compact singularities from the solutions).

On a more computational side, I will present an idea that D. Eelbode and I had during his stay at Chapman. The goal is to construct the syzygies for the Dirac system in several vector variables. I already touched upon some of the possible techniques to computer those syzygies in my previous talks. This time, I will show how to use the computer algebra software Singular and the structure of the super Lie algebra osp(1|2) to get the (minimal) free resolution of the system, in just a few command lines.

### Tuesday, April 17, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Jeff Krause, Shodor Foundation

*Title:* Preparing for the Interdisciplinary Challenges of Post-Genomic Biology:

*Abstract:* Biology is in a state of flux. The tremendous changes are due to rapidly advancing experimental technologies that are producing previously unheard of quantities of data. Storing these data and extracting information from them have caused an explosion in the field of biomedical informatics. A further quantitative emphasis results from the fact that these new experimental technologies are able to make many measurements from complex multi-scale biological systems in a single experiment. Our ability to effectively validate these data and hypothesize with them is dependent on our ability to produce predictive quantitative models of these systems. While the past decade has been truly exciting for biologists and others drawn to the field, it has resulted in a dire need for curricular reform in the biological sciences in order to prepare students for careers in post-genomic biology. In my talk I will survey the quantitative and computational landscape of contemporary biology, and I will discuss various organizations, programs and resources that have been developed in support of efforts to integrate quantitative and computational training into the undergraduate biology curriculum.

### Tuesday, April 8, 2008, 1:30-2:30 pm, BK 207

*Speaker:* David Eelbode (University of Ghent, Belgium)

*Title:* Higher spin operators in Clifford analysis:

*Abstract:* In this lecture, we will explain how one can fully exploit the framework of Clifford algebras and Clifford analysis in order to construct a function theory for higher spin Dirac operators. These are to be seen as far-reaching generalizations of the classical Dirac operator, possibly describing (exotic) elementary particles in higher dimensions. We will carefully explain how to describe the underlying invariance with respect to the underlying Lie algebra so(m), and how to relates the associated theory to Clifford analysis in several vector variables.

This work is done in collaboration with my PhD-student Liesbet Van de Voorde

### Thursday, April 3, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Cyril Rakovski, Department of Preventive Medicine, Division of Biostatistics, Keck School of Medicine, USC

*Title:* Novel statistical methods for detecting genetic association between phenotypes and multiple genes:

*Abstract:* Association methods employed for finding deleterious mutations in human populations are categorized into two broad classes with respect to the structure of the analyzed data, case-control and family-based. Case-control studies are flexible, powerful and cost-efficient approaches that possess an inherent design disadvantage, susceptibility to inflated rates of false-positive results due to unaccounted population structure and hidden relatedness. Family-based tests for association (FBATs) provide a robust alternative to case-control methods that address the above-mentioned shortcomings by conditioning on the population information. New FBAT extensions for handling multiple correlated genes and a relatedness-adjusted case-control statistical method that accounts for stratified populations are proposed and studied through extensive simulations in various settings. Our results show that in most of the analyzed scenarios the new tests attain higher power than the currently existing approaches.

### Thursday, March 6, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Hendrik de Bie, Ghent University, Belgium

*Title:* Harmonic and Clifford analysis in superspace:

*Abstract:* Superspaces are spaces with not only commuting variables but also anti-commuting variables. We will show how it is possible to extend harmonic and Clifford analysis to these superspaces by constructing a suitable representation of sl(2) and osp(1|2). Then we will use this representation to consider integration in superspaces and we will give a set of properties that uniquely determine the Berezin integral on the supersphere.

### Thursday, February 28, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Prof. Peter Jipsen, Chapman University

*Title:* Using Gentzen system techniques with existing automated theorem provers and rewriting tools

*Abstract:* Gentzen systems have been used to present many logics, such as classical logic, intuitionistic logic, modal logics, substructural logics, and their corresponding algebraic axiomatizations, in modular ways that provide new insights and often lead to effective decision procedures.
The aim of this talk is to show that Gentzen sequent calculi can be used in standard resolution theorem provers to improve their search space characteristics. This is mostly of use with (semi)lattice ordered algebras, and does not require cut-free Gentzen systems. For example it is currently not known if there is a cut-free Gentzen system for residuated Kleene algebras or residuated Kleene lattices. However if axiomatizations for these equational theories are presented in the style of Gentzen sequent rules, then a theorem prover such as Prover9
(www.prover9.org) can be quite an effective reasoning tool in these otherwise rather untractable theories.
I will also discuss how cut-free Gentzen systems can be implemented effectively using standard rewriting tools such as Maude (http://maude.cs.uiuc.edu/).

### Thursday, February 21, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Prof. Andrew Moshier, Chapman University

*Title:* Free Constructions of Directed Complete Partial Orders

*Abstract:* We develop a "generators and relations" method of constructing directed complete partial orders (dcpos), and show that the results are indeed free in the usual sense. We then continue by considering the situation where the generator set is equipped with finitary operations, showing that the free construction yields a dcpo algebra (in which the operations are Scott continuous) in the same algebraic variety as the generating structure. We apply the results to the construction of co-products, and the characterization of sub-objects, in the category of "proto-frames". This is joint work with Achim Jung and Steve Vickers of Univ. of Birmingham, UK.

### Thursday, February 7, 2008, 1:30-2:30 pm, BK 207

*Speaker:* Prof. Jack Stecher, Norwegian School of Economics and Business Administration

*Title:* Decision Making and Trade with Subjective Information