# Research

## Discrete Mathematics

### Monday, April 22, 2013

Title: An Example of an Automatic Graph of Intermediate Growth
Speaker: Dmytro Savchuk
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

The class of Cayley automatic groups was recently introduced by Kharlampovich, Khoussainov and Miasnikov as a generalization of the class of automatic groups. This class retains many nice algorithmic properties of automatic groups, but is significantly wider. In particular, it includes many groups that are not finitely presented. One of the open questions about these groups is whether there is a Cayley automatic group of intermediate growth, or, equivalently, whether there is a group of intermediate growth whose Cayley graph is automatic. We give the first example of a $$4$$-regular in finite automatic graph of intermediate growth. It is constructed as a Schreier graph of a group generated by $$3$$-state Mealy automaton. This is a joint work with Alexei Miasnikov.

### Monday, April 15, 2013

No seminar this week.

### Monday, April 8, 2013

Title: Boolean Algebras and Point-Free Topology
Speaker: Joseph Van Name
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

In this talk, I shall relate Boolean algebras with point-free topology. In point-free topology, one studies objects called frames which are complete lattices that look and behave like the lattice of all open sets in a topological space. Most notions from point-set topology can be naturally generalized to frames including continuity, most higher separation axioms, compactness and variants of compactness, uniform spaces, zero-dimensional spaces, etc. A Boolean admissibility system is a Boolean algebra along with extra structure that gives the notion which least upper bounds are important. I will give a duality between Boolean admissibility systems and zero-dimensional frames and some consequences of this duality. Furthermore, I shall characterize the Boolean admissibility systems which correspond to algebras of sets and zero-dimensional spaces in terms of strong distributivity properties.

### Monday, April 1, 2013

Title: Reverberation in Neural Circuits: Spinal Segmental Networks
Speaker: Apurva Bhatty
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

Continuing my previous talk on biological neural networks and circuits in the brain, I will discuss two motifs in neural circuits in the spinal cord at different segmental levels. The combinatorics of these two circuit motifs lead to complex interactions and reverberant circuits. Physics-based simulation of the motor control exerted by these discrete circuits in a simple graph model shows how learning can occur from Hebbian interaction at the synapses: a type of probabilistic/Bayesian learning in discrete circuits. One possible mechanism for physiologic and disease conditions, such as tremor in Parkinsons, is proposed based upon this model along with a possible explanation for how DBS (deep brain stimulation) may affect these circuits.

### Monday, March 25, 2013

Title: An Algorithm for DNA Rearrangement Pathways
Speaker: Maja Milosevic
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

We describe an algorithm which, given the DNA sequence of a ciliate micronucleus, outputs the distinct pathways of gene rearrangement. The rearrangement process is described by a graph with $$4$$-valent rigid vertices called an assembly graph. Each vertex of an assembly graph corresponds to a special sequence of DNA called a pointer, which is a recombination site that occurs at least twice in the DNA sequence. The recombination of the DNA corresponds to the smoothing of a vertex. The algorithm considers four different criteria to determine whether a vertex is smoothed: the sequence length, stress and strain, the intrinsic bendability, and possible presence of enzymes. These properties are obtained by empirical data. Edges are assigned weights based on these properties. The weighted edges are then used to rank the vertices. If a vertex has a high enough rank, that vertex is smoothed. The sequences of smoothings correspond to possible pathways of the DNA rearrangement.

### Monday, March 18, 2013

Title: New Characterizations of DNA Codewords and Their Applications
Speaker: Daniela Genova, University of North Florida
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

DNA codewords arose in the attempt to avoid unwanted hybridization of DNA strands for DNA based computation. Given a set of constraints, generating a large set of DNA strands that satisfy the constraints is an important problem in DNA computing. On the other hand, forbidding and enforcing systems (fe-systems) is a model of computation that defines classes of languages based on two sets of constraints. We establish a connection between these two areas of research in natural computing by characterizing a variety of DNA codes by fe-systems and showing how fe-systems can be used for generating good DNA code words. This talk is based on a recent paper with K. Mahalingam.

SPRING BREAK

### Monday, March 4, 2013

Title: Two-Dimensional Languages and Cellular Automata
Speaker: Egor Dolzhenko
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

The time evolution of a bi-infinite configuration of a one-dimensional cellular automaton can be visualized as a half-plane array of symbols. The set of rectangular blocks extracted from such arrays forms a two-dimensional picture language. These picture languages are called factorial-local if they can be specified by a finite set of blocks. We call a cellular automaton factorial-local if it gives rise to a factorial-local picture language. We show that the class of factorial-local cellular automata is properly included in the class of regular cellular automata and that it contains nilpotent cellular automata. We give a characterization of factorial-local cellular automata and show that it is decidable whether a given set of blocks defines a factorial-local cellular automaton with a given radius (however the general membership problem for this class of cellular automata is undecidable). A biological application of cellular automata will be presented at the end of the talk.

### Monday, February 25, 2013

No seminar this week.

### Monday, February 11, 2013

Title: Introduction to non-triviality of Gauss words
Speaker: (Joint work with Andrew Gibson)
Noboru Ito, Waseda University
Japan
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

In this talk, first we explain Gauss words that are used to represent knots in space. Second, we consider an important fact on Gauss words. Using an idea of Knot theory, we give another proof of the fact that was open until recently. This talk will be for a general audience.

### Monday, January 28, 2013

Title: Sum Formula of Multiple Hurwitz Zeta Values and Euler Sums
Speaker: Jianqiang Zhao, Eckerd College
Time: 3:05pm‐3:55pm
Place: CMC 108

#### Abstract

Multiple Hurwitz zeta values are iterated multiple variable generalizations of Hurwitz zeta value and alternating Euler sums are multiple variable generalizations of (alternating) Riemann zeta values. The number of variables is called the depththe, sum of the integer variables, the weight. In this talk we consider closed formulas of the these values with fixed weight and depth. In order to derive such formulas we are led naturally to the study of some iterated systems of differential equations whose solutions are discovered with the aid of Maple, a computer algebra system.