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ON THE MATHEMATICS OF SELF-ASSEMBLY
by
Dustin Reishus
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(COMPUTER SCIENCE)
December 2009
Copyright 2009 Dustin Reishus
Object Description
| Title | On the mathematics of self-assembly |
| Author | Reishus, Dustin |
| Author email | reishus@usc.edu; reishus@gmail.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2009-06-22 |
| Date submitted | 2009 |
| Restricted until | Unrestricted |
| Date published | 2009-11-19 |
| Advisor (committee chair) | Adleman, Leonard |
| Advisor (committee member) |
Huang, Ming-deh Rothemund, Paul Finkel, Steven |
| Abstract | Self-assembly is the ubiquitous process by which simple objects come together under simple rules to form more complex objects. Self-assembly occurs in nature to produce structures of extraordinary complexity. In the future, it may be possible to harness the power of self-assembly to manufacture useful devices in enormous quantities at little cost. In order to do so, it would be valuable to have a deep understanding of self-assembly, at both theoretical and practical levels.; I first describe experimental work with DNA self-assembly. DNA is an ideal substance to use in experimental self-assembly: It has well-understood structure; it has readily-available tools to synthesize, manipulate, and visualize it; and it has "programmable" interactions with other molecules of DNA. I describe two self-assembling DNA complexes that can further self-assemble into regular lattices.; Mathematical models of self-assembly have been created to aid in the analysis of the power and limits of self-assembly. I explore decidability questions in a mathematical model of self-assembly known as the tile assembly model. I prove the undecidability of distinguishing self-assembling systems in which infinite structures can be assembled from systems in which only finite structures can be assembled.; Many self-assembly processes are rooted in chemistry. The event-systems model generalizes the classical theory of chemical thermodynamics and places the kinetic theory of chemical reactions on a firm mathematical foundation. I prove that many of the expectations acquired through empirical study are warranted.; Finally, I use the event-systems model to explore questions in pure mathematics. The atomic hypothesis in chemistry (the theory that every substance is composed of a unique set of atoms) is analogous to the fundamental theorem of arithmetic in mathematics (the theory that every natural number is the product of a unique set of primes). I exploit this analogy by creating event-systems in which the basic components are natural numbers that can "react" through multiplication. Important thermodynamic properties such as temperature and pressure have purely mathematical implications in these systems. In particular, the pressure at equilibrium is the Riemann zeta function's value of the temperature of the system. |
| Keyword | self-assembly; DNA self-assembly; models of self-assembly; tile assembly model; event-systems; Riemann zeta function |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m2747 |
| Rights | Reishus, Dustin |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Reishus-3175 |
| Archival file | uscthesesreloadpub_Volume56/etd-Reishus-3175.pdf |
Description
| Title | Page 1 |
| Full text | ON THE MATHEMATICS OF SELF-ASSEMBLY by Dustin Reishus A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) December 2009 Copyright 2009 Dustin Reishus |
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