Abstracts

Joe Song
(Department of Computer Science, NMSU)
LOGICAL NETWORKS FOR MODELING GENE REGULATORY INTERACTIONS

Abstract: Modeling dynamical interactions among genes and environment can lead to a quantitative understanding of mechanisms in cellular processes, such as transcription regulation, metabolism, and disease pathways. Previous studies of gene regulatory networks (GRN) have established Boolean networks as dynamical models for GRNs. The Boolean network model regards every gene in a GRN as a Boolean variable whose state is a Boolean function of other genes. Current reconstruction algorithms for Boolean networks largely ignore the statistical significance of interactions. By generalization of Boolean networks to logical networks, the computational problem of identifying logical network models to account for temporal dependencies among interacting genes and environmental stimuli from high-throughput transcriptomic data is addressed. A logical network construction algorithm was developed that uses the statistical significance as a criterion for network selection to avoid false interactions arising from pure chance. Using temporal gene expression data collected from the brains of alcohol-treated mice in an analysis of the molecular response to alcohol, this algorithm identified several genes from a major neuronal pathway as putative components of the alcohol response mechanism. Three of these genes have known specific associations with alcohol response as reported in the literature. Several other potentially related were also highlighted, in agreement with independent results from literature mining. These genes may play a role in the response to alcohol. Additional, previously-unknown interactions were discovered in the logical network that, subject to biological verification, may offer new clues in the search for the elusive molecular mechanisms of alcoholism.

Luc Longpre
(Department of Computer Science, UTEP)
MEASURING PRIVACY LOSS IN STATISTICAL DATABASES

Protection of privacy in databases has become of increasing importance. While a number of techniques have been proposed to query databases while preserving privacy of individual records in the database, very little is done to define a measure on how much privacy is lost after statistical releases. We suggest a definition based on information theory. Intuitively, the privacy loss is proportional to how much the descriptional complexity of a record decreases relative to the statistical release. There are some problems with this basic definition and we suggest ways to address these problems.

Jonathan Cook
(Department of Computer Science, NMSU)
RUNTIME MONITORING AND DYNAMIC SOFTWARE UPDATING

Runtime monitoring is the task of observing software system during its execution in order to learn more about its behavior, to ensure that it is upholding certain desired properties, to detect anomalous behavior, and to possibly prevent failures from happening. Dynamic software updating is the task of "on-the-fly" loading and using new versions of software components in a continuously running system. Both tasks require an underlying framework for manipulating and managing run-time software, and this talk will discuss our research in both runtime monitoring and software updating, as well as our work in building frameworks that enable these tasks.

Luis David Lopez Gutierrez and Martine Ceberio
(Department of Computer Science, UTEP)
CSP-DRIVEN SCHEDULING ALGORITHM FOR GLOBAL COMPUTING ENVIRONMENTS

Global Computing (GC) is a emerging computing paradigm, typically consisting of one server and hundreds or thousands of heterogeneous non-dedicated hosts distributed over the internet, that donate their computational resources to scientific projects s.a.: Docking@home, Seti@home, Folding@home.

By dividing the whole program into smaller tasks, each of which being executed by a given host, this paradigm enables the distributed execution of intensive applications, consisting of millions of instructions. In an ideal world, each host would be reliable and return a valid value to the executed tasks. Unfortunately, external sources of error, s. a.: hardware malfunction, incorrect software modification, or malicious attacks, are responsible for invalid results returned by hosts.

One common solution to this problem is the strategy of replication of tasks, as proposed by Taufer et al. A problem that comes with this approach is the scheduling of these tasks, in such a way that they are appropriately distributed to similar hosts, and that it is efficient enough to reduce the number of incorrect global results.

In this work, we propose to use a CSP-driven scheduling algorithm to address the problem of scheduling tasks in a GC environment. Our algorithm guarantees an appropriate distribution of tasks to similar hosts, and experiments show that we also allow to maintain a reasonable number of incorrect results, as compared to more traditional ad-hoc approaches.

In this talk, we will present our CSP approach along with our preliminary results for both fixed-size and variable-size tasks.

Hing Leung
(Department of Computer Science, NMSU)
DESCRIPTIONAL COMPEXITY OF REGULAR LANGUAGES

A recent trend in the research of formal languages and automata is the study of the tradeoff in descriptional complexity of formal objects. Finite automata and regular languages are the most fundamental tool and concept in computer science. There have been lots of research done in descriptional complexity of finite automata and regular languages. Yet, there are still quite a few challenging and significant open problem remaining.

We'll discuss some recent works on the descriptional complexity questions about regular expressions, finite automata with limited nondeterminism, and two-way finite automata.

Son Tran
(Department of Computer Science, NMSU)
REASONING ABOUT ACTION AND CHANGE: RESEARCH DIRECTIONS AT NMSU

In this presentation I will overview the current state of the art in the field of knowledge representation, with specific focus on the problem of reasoning about action and change. I will provide specific discussion of current research directions in this area that we are pursuing at NMSU. In particular, I will discuss the use of approximation to efficiently handle forward planning in presence of incomplete knowledge (and sensing actions), as well as performing regression planning with partial knowledge. I will also highlight some recent results accomplished by applying parallel processing techniques to the problem of computing plans.

Tanja Magoc, Julio Urenda, Vladik Kreinovich
(Department of Computer Science, UTEP)
ALGORITHMIC RANDOMNESS, GENERAL NOTION OF BOUNDEDNESS, AND CATEGORY THEORY: PRELIMINARY RESULTS

Clearly, real numbers 1.0 or 0.5 or e (2.781828...) are not random, while numbers produced by real random physical processes are random. This very clear distinction between random and non-random objects has been known for several centuries, but it is very difficult to describe in precise mathematical terms. The formal definition was only given in the 20 century, via the work on von Mises, Kolmogorov, Martin-Lof and others. One known description is a description via Kolmogorov complexity. The original description is not in complete agreement with our intuition because according to this definition, a sequence of 1,000,000 0s followed by a 1 is random.

Leonid Levin used Kolmogorov complexity to provide a more intuitive definition. Crudely speaking, if we have a property that few objects satisfy, then it is not very probable that a random object has this property. So, if we have a sequence of such properties, rarer and rarer, then there is a threshold after which "random" ("typical") objects cannot satisfy one of them. For the properties "start with n zeros", we conclude that there is an integer N such that a truly random sequence cannot start with N zeros.

In this talk, we show that this advanced and difficult-to-described definition can be actually viewed as a particular case of something very natural: the general notion of boundedness. This notion – like many other general notions - is best formulated in terms of a very general language called category theory. In this language, we have objects and mapping (called "morphisms"). For example, in the category Set objects are sets, and mapping are arbitrary functions. In the category Ord, objects are ordered sets, and mapping are monotonic functions. We can define a subset S of an object A to be bounded if for every morphism f from A to real numbers R the range f(S) is bounded. It turns out that for many categories, this notion of boundedness is equivalent to a natural one: for Set, S is bounded if and only it is finite; for linearly ordered sets, f is bounded if and only if it is bounded in the normal sense; for continuous mappings, bounded means compact etc.

We show that if instead of the category of ALL sets and functions, we consider only definable sets and functions, then boundedness is exactly randomness (typicalness) in the sense of the above definition. Thus, the seemingly complex notion of randomness is, in fact, very mathematically natural.

Tiziana Giorgi and Robert Smits
(Department of Mathematical Sciences, NMSU)
EIGENVALUE ESTIMATES FOR THE ROBIN PROBLEM

In our joint research we recently looked at eigenvalue problems which arise from the linearization of interface models in superconductivity and other applied areas. We derived new upper and lower bounds for the Robin problem which have physical relevance. We plan to continue our study of eigenvalue problems with applications in Materials Science.

David Andrs
(Department of Mathematics, UTEP):
CONTROLLED DATA COMPRESSION USING ADAPTIVE FINITE ELEMENTS

We are concerned with the compression of 2D and 3D data (images, measured quantities, results of MRI scans, etc.) using adaptive higher-order finite element methods (hp-FEM). The basic idea of such data compression is to approximate a function of many parameters

(the image) as accurately as possible and using as few parameters as possible. We employ the hp-FEM machinery to achieve this goal. Compared to other data compression techniques, this approach offers a strong mathematical background and an exact control over the quality of approximation (using norms in the appropriate function spaces). We expect to present comparisons to other image compression techniques such as JPEG.

Pavel Solin
(Department of Mathematical Sciences, UTEP)
FEM OF THE FUTURE AND MODULAR FINITE ELEMENT SYSTEM HERMES

We will describe briefly adaptive higher-order finite element methods (hp-FEM) and explain why we expect that finite element analysis in engineering and science will go into this direction in the next years. A large obstacle on the way is considerable mathematical and programming difficulty and lack of software. Besides developing the hp-FEM on the theoretical level, it is one of our major goals to build such software. We describe our modular finite element system HERMES which we hope to make available on-line in summer or fall 2007. Open problems/outlook/future plans will be mentioned.

Maria Christina Mariani
(Department of Mathematical Sciences, NMSU)
EXTREME EVENTS IN FINANCIAL MARKETS

Over the past two decades, the complexity of international finance has grown enormously with the development of new markets and instruments for transferring risks. This growth in complexity has been accompanied by an expanded role for mathematical models to value derivative securities and to measure their risks. This will be undertaken through two specific problems in the mathematics of risk management:
1. The analysis of asset-price dynamics in models that capture the possibility of sudden, large changes in prices --- i.e., "jumps".

2. The development and application of tools from mathematical physics to analyze market dynamics leading to a "crash."

Ising type models, Normalized Truncated Levy distributions, and the relation between intermittence and scale invariance will be discussed.

Ronald H. W. Hoppe
(Department of Mathematics, University of Houston;
Institute of Mathematics, University of Augsburg, Germany)
MODELING AND SIMULATION OF PIEZOELECTRICALLY AGITATED ACOUSTIC STREAMING ON MICROFLUIDIC BIOCHIPS

Biochips, of the microarray type, are fast becoming the default tool for combinatorial chemical and biological analysis in environmental and medical studies. Programmable biochips are miniaturized biochemical labs that are physically and/or electronically controllable. The technology combines digital photolithography, microfluidics and chemistry. The precise positioning of the samples (e.g., DNA solutes or proteins) on the surface of the chip in picoliter to nano liter volumes can be done either by means of external forces (active devices) or by specific geometric patterns (passive devices). The active devices which will be considered here are nano liter fluidic biochips where the core of the technology are nano pumps featuring surface acoustic waves generated by electric pulses of high frequency. These waves propagate like a miniaturized earthquake (nanoscale earthquake), enter the fluid filled channels on top of the chip and cause an acoustic streaming in the fluid which provides the transport of the samples. The mathematical model represents a multiphysics problem consisting of the piezoelectric equations coupled with multiscale compressible Navier-Stokes equations that have to be treated by an appropriate homogenization approach. We discuss the modeling approach and present algorithmic tools for numerical simulations as well as visualizations of simulation results.

John Harding
(Department of Mathematical Sciences, NMSU)
LATTICES AND THEIR APPLICATIONS

A lattice is a partially ordered set where any two elements have a greatest lower bound and a least upper bound. Examples of lattices include the natural numbers under the usual ordering, and the power set of a set under the partial ordering of set inclusion.

The utility of such a simple concept of a lattice is that lattices appear naturally in nearly all branches of mathematics. This applies to analysis, where one considers vector lattices; algebra, where one considers lattices of normal subgroups and submodules; logic, where the propositions of a logic form a lattice under the ordering of implication; geometry, where the points, lines, and planes of an affine space form a lattice; and quantum mechanics, where the closed subspaces of a Hilbert space form a lattice under set inclusion.
This list could be extended a hundred fold.

In this brief talk, I'll try to present some of the basic ideas of lattices, and indicate some of the types of problems that we study in this area at NMSU.

Joel Lucero-Bryan
(Department of Mathematical Sciences, NMSU)
MODAL LOGIC AND TOPOLOGY

Modal logic is an extension of classical logic, which, among other things, provides formalization of such fundamental concepts as necessity, possibility, knowledge, belief, and others. It has wide applications in many areas, including computer science, artificial intelligence, philosophy, linguistics, and others. The standard semantics of modal logic is the relational semantics, introduced by Kripke and others in the late 50ies and early 60ies. A new trend in modal logic is topological semantics, which in some good cases, is a generalization of the relational semantics. In this talk I'll discuss some of the work done at NMSU on topological semantics of modal logic.

Hung T. Nguyen
(Department of Mathematical Sciences, NMSU):
ON MATHEMATICAL FOUNDATIONS FOR THE ANALYSIS OF STATISTICAL COARSE DATA

It is well-known that while statistical theory can be formulated in its most general form, specific domains of applications require a careful investigation. This is due mainly to the types of observed data. Thus, for example, survival analysis differs from traditional and standard statistical analysis by the fact that observed data are censored; inference in bioinformatics faces the problem of partially observed or hidden data. These types of imprecise, incomplete data are termed coarse data, and require the development of new statistical procedures.

In this talk we will consider the types of statistical data which are sets and fuzzy sets. To analyze such data, we need first to model mathematically the observed processes. Thus we will present our current research on developing a general and rigorous theory of random sets and upper semicontinuous random functions together with their probabilistic distribution theory.

Sandor Jenei
(Department of Mathematical Sciences, NMSU;
Institute of Mathematics and Informatics, University of Pecs, Hungary)
GAME THEORETIC APPLICATIONS OF MATHEMATICAL FUZZY LOGICS

Let e be a natural number. Pinocchio thinks of a natural number s between 1 and 10.000.000. The Questioner asks questions like "Is the number in X?", where X is a subset of the search space. Pinocchio answers the question but he can lie up to e-times, if he wants to. What is the/an optimal strategy for the Questioner to find the secret number s?

This problem was first posed by A. Renyi, a few years after him independently by S. Ulam, and is called the Rényi-Ulam game. This problem was motivated by the need of finding so-called adaptive codes in coding theory. Many mathematicians have worked on this topic in the last few decades, and by now this problem has been solved for all e. Still some work is to be done to apply only "simple" questions in the strategy of the Questioner.

Following D. Mundici we point out that there is a strong link between this game and the famous Lukasiewicz logic. A similar game interpretation is given for the Basic Logic1 of P. Hajek (due to Cicalese and Mundici) and for the Product Logic2 (due to S. Jenei and F. Montagna).

1. BL is the logic of all continuous t-norms.

2. Product Logic is the logic of the product t-norm.

Jerry Lodder
(Department of Mathematical Sciences, NMSU)
DIFFERENTIABLE MANIFOLDS, 2-ASSOCIATIVE ALGEBRAS, AND ROOTED TREES

Beginning with the de Rham complex of differential forms on a manifold, we examine a new cohomology theory that includes all tensors from differential geometry. The Riemann curvature tensor, studied heavily in the Poincar\'e Conjecture, has coboundary that reduces to the derivative of sectional curvature, thus imparting a geometric meaning to the new cohomology theory. The algebraic setting for derived functors in this theory is that of 2-associative algebras and their quotients. The free 2-associative algebra can be described in terms of grafting operations on rooted n-ary trees, and these in turn index the summands in the chain complex for the new cohomology theories.