DOCSLIB.ORG

Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing Western University Scholarship@Western Electronic Thesis and Dissertation Repository 12-1-2017 11:00 AM Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing Amirhossein Simjour The University of Western Ontario Supervisor Prof. Lila Kari The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Amirhossein Simjour 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Other Computer Sciences Commons, and the Theory and Algorithms Commons Recommended Citation Simjour, Amirhossein, "Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing" (2017). Electronic Thesis and Dissertation Repository. 5137. https://ir.lib.uwo.ca/etd/5137 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick comple- mentarity of DNA sequences, without external direction. In this research we study the theoretical power of different mathematical models of self- assembly systems. The presented models are proposed to reduce the gap between self-assembly models and models that have been used in existing theoretical researches such as language theory and robotics. We propose self-assembly (SA) hypergraph automata as an automata- theoretic model for patterned self-assembly. We investigate the computational power of SA- hypergraph automata and show that for every recognizable picture language, there exists an SA- hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. We also study complex self-assembly models and investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of Smart Tiles, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes. Keywords: DNA based self-assembly, self-assembly of tiles, hypergraph automata, smart tiles i Co-Authorship Statement This thesis consists of three research articles that are published or are submitted. For all the articles, the authors names are listed alphabetically. The contribution of each of the co-authors are listed below. The article in chapter three ”Hypergraph Automata: a Theoretical Model for Patterned Self- assembly.” by Lila Kari, Steffen Kopecki, Amirhossein Simjour is published in International Journal of Foundations of Computer Science 25(4): 419-440 (2014). LK- research idea, co-writing proofs of theorems; SK- co-writing proofs of theorems, rewriting of the theorem 3.4.1, manuscript writing and editing; AS- research idea, first manuscript draft, proof of the theorems. The article in chapter four ”Simplifying the Role of Signals in Tile Self-assembly.” by Lila Kari, Amirhossein Simjour is submitted and is under review. LK- research idea, co-writing proofs of theorems, manuscript writing and editing; AS- research idea, first manuscript draft, proof of the theorems. The article in chapter five ”Smart Tile Self-Assembly and Replication.” by Lila Kari, Amirhossein Simjour is published in Fundamenta Informaticae 154(1-4): 239-260 (2017) . LK- research idea (specifically started the idea of definition of tiles with computational power), co-writing proofs of theorems, manuscript writing and editing; AS- research idea, first manuscript draft, proof of the theorem. ii Acknowledgements I am glad to acknowledge all of those who played a role to complete this thesis, whether large or small. I would like to take this opportunity to express how truly grateful I am to my supervisor, Dr. Lila Kari for her continuous commitment and support she has shown me throughout my study. To my sister, Narges who supported me in Canada and played the role of my entire family, thank you. I would like to thank my parents for all the support from thousands of miles away. Thank you for being there exactly when I needed you. Last but not the least, thanks to my wife Mahboubeh who have given me spirit and support in completing my thesis. THANK YOU! iii Contents Abstract i Co-Authorship Statement ii Acknowlegements iii List of Figures vii List of Tables x List of Appendices xi 1 Introduction 1 1.1 Hypergraph Automata: A Theoretical Model for Patterned Self-assembly . 2 1.2 The Computational Power of Glue-deactivating Signals in Tile Self-assembly . 3 1.3 Smart Tiles and Replication . 3 2 Literature review 5 2.1 Preliminaries . 6 2.1.1 Mathematical Description . 8 2.1.2 Computational Power . 12 2.2 Error in Self-assembly . 15 2.3 DNA-based Nanorobotics . 18 2.4 2D Grammars . 21 Bibliography 25 iv 3 Hypergraph Automata: A Theoretical Model for Patterned Self-assembly 33 3.1 Introduction . 33 3.2 Preliminaries . 35 3.3 Hypergraph Automata . 38 3.4 Hypergraph Automata for Picture Languages . 45 3.5 Conclusion . 56 Bibliography 57 4 Simplifying the Role of Signals in Tile Self-assembly 60 4.1 Introduction . 60 4.2 The Detachable Tile Assembly Model . 62 4.2.1 Formal Definition of DTAM . 64 4.2.2 Transitions . 66 4.3 Turing Universality . 72 4.4 Self-assembly of Thin Rectangles . 79 4.4.1 Informal description of the DTAM tile assembly system . 79 4.4.2 Detailed description of the DTAM tile assembly system . 82 4.4.3 DTAM tile complexity and waste analysis . 95 4.5 Conclusions . 96 Bibliography 96 4.6 Addendum . 99 5 Smart Tile Self-Assembly and Replication 101 5.1 Introduction . 101 5.2 Smart Tile Assembly . 103 5.2.1 Basic Definitions . 103 5.2.2 The Smart Tile Assembly Model . 106 5.3 A Smart Tile Assembly System that Replicates L-Convex Shapes . 113 5.4 Discussion and Future Work . 126 v Bibliography 127 6 Conclusion 130 A Appendix: Examples of SA-Hypergraph Automata 132 Curriculum Vitae 137 vi List of Figures 2.1 DAO and DAE tiles . 8 2.2 An example of a tile assembly system with a seed . 11 2.3 An example of flipped tiles . 17 3.1 An example of Wang tile system . 37 3.2 The self-assembly of a single coloured pattern . 38 3.3 Transition in hypergraph automata . 41 3.4 Underlying graph of the SA-hypergraph automaton that constructs a coloured pattern . 42 3.5 Step by step construction of a picture graph . 44 3.6 Hyperedges of a hypergraph automaton A that recognizes a recognizable pic- ture language L .................................. 47 3.7 Want tile system that accepts the same language as an SA-hypergraph automaton 51 4.1 A detachable tile in the DTAM . 64 4.2 An example of a general configuration and the assembly-graph associated to it . 68 4.3 An example of an attachment transition . 70 4.4 Simulation of a Turing machine using a deterministic zig-zag tile assembly system at temperature 2 . 74 4.5 The original tiles from the deterministic zig-zag tile assembly model at temper- ature 2, and their corresponding gadgets made out of SDTAM tiles . 75 4.6 The tiles that are used in the bit reader part of the gadgets . 76 4.7 The idea of the steps that replace an M × (M! + 2C) rectangle by an (M + 1) × ((M + 1)! + 2C) rectangle . 80 4.8 The use of the mirror rectangle in preventing unexpected attachments . 81 vii 4.9 The attachment of two rectangles and start of the column replacement . 83 4.10 A thin rectangle that is the input of Step 1 of the construction, and the glues on the tiles of its bottom edge . 84 4.11 The tile types that are used in the construction of a thin rectangle and their respective glues . 85 4.12 A detailed view of Step 2, and the attachment of the first column in Step 3 of the construction of a thin rectangle . 87 4.13 Step 3 (expansion), and Step 4 (reattachment) of the construction of a thin rectangle. Part 1 . 88 4.14 Step 3 (expansion), and Step 4 (reattachment) of the construction of a thin rectangle. Part 2 . 89 4.15 The tiles that are involved in the attachment of the left and right parts of the rectangles . 95 5.1 An example of an attachment transition . 108 5.2 A smart-TAM system at temperature 1 with a single tile type that can assemble a final configuration consisting of a row of three tiles . 112 5.3 An example of an L-convex supertile . 114 5.4 An example of an L-convex supertile with the border glues needed for its repli- cation by the smart-TAM system . 116 5.5 The general idea of the replication process of an L-convex supertile of a given shape, by a smart tile self-assembly system . 117 5.6 The bijective mapping between the glues on the border of the original supertile and the glues on the border of the quarter-rectangle . 117 5.7 The result of Step 1: Filling all four quarter-rectangles around the original supertile . 118 5.8 The rectangular hole that is obtained at the end of Step 2 . 119 5.9 The tile set Θborder-ne = fT1;:::; T6g, and Θreplica = fTreplicag .
Load more

Recommended publications
  • Computational Design Framework 3D Graphic Statics
    Computational Design Framework for 3D Graphic Statics 3D Graphic for Computational Design Framework Computational Design Framework for 3D Graphic Statics Juney Lee Juney Lee Juney ETH Zurich • PhD Dissertation No. 25526 Diss. ETH No. 25526 DOI: 10.3929/ethz-b-000331210 Computational Design Framework for 3D Graphic Statics A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Juney Lee 2015 ITA Architecture & Technology Fellow Supervisor Prof. Dr. Philippe Block Technical supervisor Dr. Tom Van Mele Co-advisors Hon. D.Sc. William F. Baker Prof. Allan McRobie PhD defended on October 10th, 2018 Degree confirmed at the Department Conference on December 5th, 2018 Printed in June, 2019 For my parents who made me, for Dahmi who raised me, and for Seung-Jin who completed me. Acknowledgements I am forever indebted to the Block Research Group, which is truly greater than the sum of its diverse and talented individuals. The camaraderie, respect and support that every member of the group has for one another were paramount to the completion of this dissertation. I sincerely thank the current and former members of the group who accompanied me through this journey from close and afar. I will cherish the friendships I have made within the group for the rest of my life. I am tremendously thankful to the two leaders of the Block Research Group, Prof. Dr. Philippe Block and Dr. Tom Van Mele. This dissertation would not have been possible without my advisor Prof. Block and his relentless enthusiasm, creative vision and inspiring mentorship.
    [Show full text]
  • Bottom-Up Self-Assembly Based on DNA Nanotechnology
    nanomaterials Review Bottom-Up Self-Assembly Based on DNA Nanotechnology 1, 1, 1 1 1,2,3, Xuehui Yan y, Shujing Huang y, Yong Wang , Yuanyuan Tang and Ye Tian * 1 College of Engineering and Applied Sciences, State Key Laboratory of Analytical Chemistry for Life Science, Nanjing University, Nanjing 210023, China; [email protected] (X.Y.); [email protected] (S.H.); [email protected] (Y.W.); [email protected] (Y.T.) 2 Shenzhen Research Institute of Nanjing University, Shenzhen 518000, China 3 Chemistry and Biomedicine Innovation Center, Nanjing University, Nanjing 210023, China * Correspondence: [email protected] These authors contributed equally to this work. y Received: 9 September 2020; Accepted: 12 October 2020; Published: 16 October 2020 Abstract: Manipulating materials at the atomic scale is one of the goals of the development of chemistry and materials science, as it provides the possibility to customize material properties; however, it still remains a huge challenge. Using DNA self-assembly, materials can be controlled at the nano scale to achieve atomic- or nano-scaled fabrication. The programmability and addressability of DNA molecules can be applied to realize the self-assembly of materials from the bottom-up, which is called DNA nanotechnology. DNA nanotechnology does not focus on the biological functions of DNA molecules, but combines them into motifs, and then assembles these motifs to form ordered two-dimensional (2D) or three-dimensional (3D) lattices. These lattices can serve as general templates to regulate the assembly of guest materials. In this review, we introduce three typical DNA self-assembly strategies in this field and highlight the significant progress of each.
    [Show full text]
  • Rational Design of DNA Nanoarchitectures Udo Feldkamp* and Christof M
    Reviews U. Feldkamp and C. M. Niemeyer DOI: 10.1002/anie.200502358 Nanoscience Rational Design of DNA Nanoarchitectures Udo Feldkamp* and Christof M. Niemeyer* Keywords: DNA · materials science · nanostructures · self-assembly · supramolecular chemistry Angewandte Chemie 1856 www.angewandte.org 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2006, 45, 1856 – 1876 Angewandte DNANanoarchitectures Chemie DNA has many physical and chemical properties that make it a From the Contents powerful material for molecular constructions at the nanometer length scale. In particular, its ability to form duplexes and other 1. Introduction 1857 secondary structures through predictable nucleotide-sequence- 2. General Considerations of DNA- directed hybridization allows for the design of programmable Sequence Design 1858 structural motifs which can self-assemble to form large supra- molecular arrays, scaffolds, and even mechanical and logical 3. One-Dimensional DNA Strands for nanodevices. Despite the large variety of structural motifs used Assembly and Immobilization of Non- Nucleic Acid Compounds 1859 as building blocks in the programmed assembly of supra- molecular DNA nanoarchitectures, the various modules share 4. Design and Assembly of DNA Motifs 1860 underlying principles in terms of the design of their hierarchical configuration and the implemented nucleotide sequences. This 5. Three-Dimensional Structures from DNA 1866 Review is intended to provide an overview of this fascinating and rapidly growing field of research from the structural design 6. Applications of DNA Nanoarchitectures 1868 point of view. 7. Conclusions and Perspectives 1872 1. Introduction combination of distinct ssDNA and dsDNA elements within an artificial DNA motif allows structural building blocks to be Biomolecules, such as proteins and nucleic acids, possess designed with tailored flexibility and rigidity.
    [Show full text]
  • Computability and Tiling Problems
    Computability and Tiling Problems Mark Richard Carney University of Leeds School of Mathematics Submitted in accordance with the requirements for the degree of Doctor of Philosophy October 2019 Intellectual Property Statement The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copy- right material and that no quotation from the thesis may be pub- lished without proper acknowledgement. The right of Mark Richard Carney to be identified as Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. c October 2019 The University of Leeds and Mark Richard Carney. i Abstract In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles S has total planar tilings, which we denote T ILE, or whether it has infinite connected but not necessarily total tilings, W T ILE (short for ‘weakly tile’). We show that both T ILE ≡m ILL ≡m W T ILE, and thereby both T ILE 1 and W T ILE are Σ1-complete. We also show that the opposite problems, :T ILE and SNT (short for ‘Strongly Not Tile’) are such that :T ILE ≡m W ELL ≡m 1 SNT and so both :T ILE and SNT are both Π1-complete.
    [Show full text]
  • How to Construct Convex Polyominoes on DNA Wang Tiles ?
    How to construct convex polyominoes on DNA Wang tiles ? F. De Carli ∗ A. Frosini ∗ S. Rinaldi ∗ L. Vuillon † 1 Introduction The goal of this article is to present a method to construct various classes of convex polyominoes using DNA Wang tiles. The construction is based on some results of [4], where the authors present a coding of convex polyominoes in terms of two dimensional languages, by means of tiling systems [12]. Adopting the same formalism we used in [4], and applying an algorithm by L. De Prophetis and S. Varricchio [11], we are able to transform the tiles of the tiling systems for convex polyominoes to labelled Wang tiles. The last step of the construction is based on a conversion from labelled Wang tiles into DNA Wang tiles [1], which gives an effective way to construct nano structures with convex polyomino shapes. The possibility of constructing polyominoes by means of DNA Wang tiles seems interesting since polyominoes are simple and important discrete structures that appear in several problems related to theoretical computer science and discrete mathematics. In the half century, since Solomon Golomb used the term in his seminal article [13], the study of polyominoes has proved a fertile topic of research. By this period in the mid-1950s, it was clearly a timely notion in discrete models, as the increasingly influential work of Neville Temperley, on problems drawn from statistical mechanics and molecular dynamics [16], and of John Hammersely, dealing with percolation [14], bear witness. More recent years have seen the treatment of numerous related problems, such as the problem of covering a polyomino by rectangles [5] or problems of tiling regions by polyominoes [2, 6].
    [Show full text]
  • Efficient Methods
    Efficient Methods for Tile-Based Synthesis and Computational Photography DISSERTATION zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften an der Universitat¨ Konstanz Fachbereich Informatik & Informationswissenschaft vorgelegt von Johannes Peter Kopf Tag der mundlichen¨ Prufung:¨ 27. Oktober 2008 1. Referent: Prof. Dr. Oliver Deussen 2. Referent: Prof. Dr. Dietmar Saupe 3. Referent: Prof. Dr. Hans-Peter Seidel Abstract This thesis presents contributions to two major topics in computer graphics. The first part de- scribes new algorithms for tile-based synthesis of blue noise point sets and solid textures; the second part describes systems for capturing, viewing, and manipulating outdoor photographs. Well distributed point sets play an important role in computer graphics, as well as many other fields, since they lie in the very foundation of any sampling technique. Many previous re- searchers have pointed out that point sets with a blue noise Fourier spectrum, i.e. the points are distributed both evenly and randomly, are desirable in many applications. In this thesis, we introduce a novel technique for rapidly generating such point sets. Through the use of Wang tiles, our technique deterministically generates infinite non-periodic patterns. Any local area may be consistently regenerated as needed. The points in each tile form a progressive sequence, enabling matching arbitrary spatially varying point densities. Recursion allows our technique to adaptively subdivide tiles where high density is required, and makes it possible to zoom into point sets by an arbitrary amount, while maintaining a constant apparent density. The technique is extremely fast (point generation is in the order of several millions of points per second) and has a compact memory footprint.
    [Show full text]
  • An Introduction to Tile-Based Self-Assembly and a Survey of Recent Results
    Nat Comput DOI 10.1007/s11047-013-9379-4 An introduction to tile-based self-assembly and a survey of recent results Matthew J. Patitz Ó Springer Science+Business Media Dordrecht 2013 Abstract We first give an introduction to the field of tile- state, spontaneously and without external guidance coa- based self-assembly, focusing primarily on theoretical lesce to form more complex structures. The process is models and their algorithmic nature. We start with a guided by only local interactions between the components, description of Winfree’s abstract Tile Assembly Model which typically follow a basic set of rules. Despite the (aTAM) and survey a series of results in that model, dis- seemingly simplistic nature of self-assembly, its power can cussing topics such as the shapes which can be built and the be harnessed to form structures of incredible complexity computations which can be performed, among many others. and intricacy. In fact, self-assembling systems abound in Next, we introduce the more experimentally realistic kinetic nature, resulting in everything from the delicate crystalline Tile Assembly Model (kTAM) and provide an overview of structure of snowflakes to many of the structurally and kTAM results, focusing especially on the kTAM’s ability to functionally varied components of biological systems. model errors and several results targeted at preventing and Beyond the purely mathematically interesting properties correcting errors. We then describe the 2-Handed Assembly of self-assembling systems, such systems have been rec- Model (2HAM), which allows entire assemblies to combine ognized as an excellent template for the fabrication of with each other in pairs (as opposed to the restriction of artificial structures on the nanoscale.
    [Show full text]
  • Wang Tile and Aperiodic Sets of Tiles
    Wang Tile and Aperiodic Sets of Tiles Chao Yang Guangzhou Discrete Mathematics Seminar, 2017 Room 416, School of Mathematics Sun Yat-Sen University, Guangzhou, China Hilbert’s Entscheidungsproblem The Entscheidungsproblem (German for "decision problem") was posed by David Hilbert in 1928. He asked for an algorithm to solve the following problem. I Input: A statement of a first-order logic (possibly with a finite number of axioms), I Output: "Yes" if the statement is universally valid (or equivalently the statement is provable from the axioms using the rules of logic), and "No" otherwise. Church-Turing Theorem Theorem (1936, Church and Turing, independently) The Entscheidungsproblem is undecidable. Wang Tile In order to study the decidability of a fragment of first-order logic, the statements with the form (8x)(9y)(8z)P(x; y; z), Hao Wang introduced the Wang Tile in 1961. Wang’s Domino Problem Given a set of Wang tiles, is it possible to tile the infinite plane with them? Hao Wang Hao Wang (Ó, 1921-1995), Chinese American philosopher, logician, mathematician. Undecidability and Aperiodic Tiling Conjecture (1961, Wang) If a finite set of Wang tiles can tile the plane, then it can tile the plane periodically. If Wang’s conjecture is true, there exists an algorithm to decide whether a given finite set of Wang tiles can tile the plane. (By Konig’s infinity lemma) Domino problem is undecidable Wang’s student, Robert Berger, gave a negative answer to the Domino Problem, by reduction from the Halting Problem. It was also Berger who coined the term "Wang Tiles".
    [Show full text]
  • University of Southampton Research Repository Eprints Soton
    University of Southampton Research Repository ePrints Soton Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk UNIVERSITY OF SOUTHAMPTON Faculty of Engineering, Science and Mathematics School of Chemistry Synthesis of oligonucleotide analogues for use in DNA nanostructures By Adeline Durand A thesis submitted for the degree of Doctor of Philosophy. March 2010 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING, SCIENCE & MATHEMATICS SCHOOL OF CHEMISTRY Doctor of Philosophy SYNTHESIS OF OLIGONUCLEOTIDE ANALOGUES FOR USE IN DNA NANOSTRUCTURES by Adeline Durand Thanks to its ability to form duplexes through selective base-pair recognition, DNA is a unique material for orderly self-assembled construction at the nanoscale. To develop a nanotechnology platform on a grid of addressable molecular building blocks using DNA node structures, DNA complexes need to be fixed onto surfaces. To fulfil this requirement on lipid membranes, phosphoramidites monomers modified with a cholesterol moiety and a spacer unit were synthesised.
    [Show full text]
  • Search Methods for Tile Sets in Patterned DNA Self-Assembly$
    Search methods for tile sets in patterned DNA self-assemblyI Mika G¨o¨os1, Tuomo Lempi¨ainen2, Eugen Czeizler3,∗, Pekka Orponen3 Department of Information and Computer Science and Helsinki Institute for Information Technology HIIT Aalto University, Finland Abstract The Pattern self-Assembly Tile set Synthesis (PATS) problem, which arises in the theory of structured DNA self-assembly, is to determine a set of coloured tiles that, starting from a bordering seed structure, self-assembles to a given rectangular colour pattern. The task of finding minimum-size tile sets is known to be NP-hard. We explore several complete and incomplete search techniques for finding minimal, or at least small, tile sets and also assess the reliability of the solutions obtained according to the kinetic Tile Assembly Model. Keywords: DNA self-assembly, tilings, Tile Assembly Model, pattern assembly, tile set synthesis, reliable self-assembly 1. Introduction Algorithmic assembly of nucleic acids (DNA and RNA) has advanced extensively in the past 30 years, from a seminal idea to the current designs and experimental implementations of complex nanostructures and nanodevices with dynamic, programmable evolution and machinelike properties. Recent developments in the field include fundamental constructions such as in vitro complex 3D pattern formation and functionalisation [3, 4], robotic designs such as mobile arms, walkers, motors [5, 6], computational primitives [7, 8], and also applications to in vivo biosensors [9] and potential drug delivery mechanisms and therapeutics [10]. Self-assembly of nucleic acids can be seen both as a form of structural nanotechnology and as a model of computation. As a computational model, one first encodes the input of a computational problem into an algorithmically designed (DNA) pattern or shape.
    [Show full text]
  • Tiling with Polyominoes, Polycubes, and Rectangles
    University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2015 Tiling with Polyominoes, Polycubes, and Rectangles Michael Saxton University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Saxton, Michael, "Tiling with Polyominoes, Polycubes, and Rectangles" (2015). Electronic Theses and Dissertations, 2004-2019. 1438. https://stars.library.ucf.edu/etd/1438 TILING WITH POLYOMINOES, POLYCUBES, AND RECTANGLES by Michael A. Saxton Jr. B.S. University of Central Florida, 2013 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Fall Term 2015 Major Professor: Michael Reid ABSTRACT In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z × Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes. ii ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor, Professor Michael Reid, who spent countless hours mentoring me over my time at the University of Central Florida.
    [Show full text]
  • Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain Gurprit Singh
    Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain Gurprit Singh To cite this version: Gurprit Singh. Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain. Sig- nal and Image Processing. Université Claude Bernard - Lyon I, 2015. English. NNT : 2015LYO10121. tel-01217082v2 HAL Id: tel-01217082 https://hal.archives-ouvertes.fr/tel-01217082v2 Submitted on 17 Jun 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thesis for Doctor of Philosophy (PhD) Sampling and Variance Analysis for Monte Carlo Integration in Spherical Domain Author: Gurprit Singh 8th September, 2015 Jury Members Prof. Jean-Michel Dischler rapporteur Dr. Nicolas Holzschuch rapporteur Prof. Michael Kazhdan examinateur Dr. Bruno Levy´ examinateur Prof. Victor Ostromoukhov (directeur) examinateur i ©2015 Gurprit Singh / Gurpreet Singh Bagga ALL RIGHTS RESERVED ii To my parents: Gurmail & Surinder iii Acknowledgements This dissertation would not have been possible without the continous support of my advisor, Prof. Victor Ostromoukhov. Through out the period of my stay at LIRIS lab Victor showed great patience and responded to my countless questions. I am grateful to Victor for motivating me to gain an in-depth knowledge of the topics I wanted to work on.
    [Show full text]