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Mathematics & Statistics

Discrete Mathematics
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Monday, September 16, 2019

Title
Speaker
Time
Place

On connection between Mealy and Moore automata via \(d\)-adic dynamics
Dima Savchuk
2:00pm–2:50pm
CMC 108

Abstract

The ring \(Z_d\) of \(d\)-adic integers has a natural interpretation as the boundary of a rooted \(d\)-ary tree. Endomorphisms of this tree are in one-to-one correspondence with 1-Lipschitz mappings from \(Z_d\) to itself as was shown by Anashin. Therefore, to study such mappings we can use the language of endomorphisms of rooted trees and, in particular, the techniques of Mealy automata. In this talk we give an explicit connection between the van der Put series of a 1-Lipschitz mapping from \(Z_d\) to itself and the structure of an associated Mealy automaton defining this mapping. This gives a way to construct Mealy automata of mappings that are defined by automatic sequences, like Thue-Morse, for example. This is a joint work with Rostislav Grigorchuk.

Monday, September 9, 2019

Title
Speaker
Time
Place

Combinatorial Algorithms for the Design of Biomolecular Nanostructures
Abdulmelik Mohammed
2:00pm–2:50pm
CMC 108

Abstract

Over the past three decades, nucleic acids have been successfully demonstrated as nanoscale building materials for the assembly of a variety of shapes, patterns, crystals and computing devices. As the scale and complexity of biomolecular nanostructures increase, the need for the algorithmic formulation of design challenges and the automation of design processes becomes more apparent.

In this talk, we present the algorithmic design of DNA and RNA nanostructures based on surface meshes as discrete approximations of target shapes. An Eulerian-tour model is used for unknotted routing of circular DNA strands on mesh skeleta and we review algorithms for such tours. We also discuss a recent approach to spanning-tree based design of 3D RNA nanostructures from single stranded RNA.