DOCSLIB.ORG

Design and Verification of Modular Components in Thermodynamic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Design and Verification of Modular Components in Thermodynamic Design and Verification of Modular Components in Thermodynamic Binding Networks By David Russell Haley Dissertation Submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Applied Mathematics in the Office of Graduate Studies of the University of California Davis Approved: David Doty, Chair Raissa D'Souza James Crutchfield Committee in Charge 2021 i Copyright © 2021 by David Russell Haley All rights reserved. This work is dedicated to everyone who has in some way supported me or encouraged me over the course of my career. This group is in total too numerous to name here, but the individuals know who they are, even though most will be modest and try to underplay their particular role. I especially thank my wife, Kelly Meyer, for her exceptional patience over the better part of a decade during which she had front-row tickets with which to observe the often trying experiences of graduate education. ii Contents Abstract . vi 1 Introduction 2 2 Thermodynamic Binding Networks Model 8 2.1 Model . .8 2.1.1 TBN . .8 2.1.2 Configuration . .8 2.1.3 Path . .9 2.1.4 Energy . 10 2.1.5 Barrier . 12 2.2 Saturated paths . 13 2.2.1 Bounds on energy change . 13 2.2.2 Saturated paths suffice . 14 2.3 Modeling bonds . 16 2.3.1 Bond-Aware Model . 16 2.3.2 Equivalence . 17 3 Engineering Systems with Programmable Energy Barriers 19 3.1 Translator cycle . 20 3.1.1 Construction . 20 3.1.2 Proof of Barrier . 30 3.2 Grid gate . 32 3.2.1 Construction . 32 3.2.2 Proof of Barrier . 33 3.2.3 Catalysis . 34 3.2.4 Autocatalysis . 39 3.3 Reducing binding site complexity for multistable TBNs . 41 3.3.1 Golfergate: Pairwise intersection 1 of state monomers . 42 iii 3.3.2 Catalysts . 44 3.3.3 Three states . 45 3.3.4 Many states . 46 p 3.3.5 Catalyst with Barrier n=2 − n ln n ................. 48 4 PSPACE-completeness of Reachability 51 4.1 Introduction . 51 4.1.1 Related work . 52 4.2 Model . 54 4.2.1 Chemical reaction networks . 54 4.2.2 Constraints on CRNs . 55 4.3 PSPACE-completeness of binary, reversible, singular, catalytic CRNs . 56 4.3.1 Definition of CRN simulation . 56 4.3.2 Construction . 58 4.3.3 Proof . 61 4.4 Singular ≤3-catalytic CRNs can simulate singular catalytic CRNs . 69 4.4.1 Simulation construction . 69 4.5 TBNs can simulate (restricted) CRNs . 74 4.5.1 Thermodynamic Binding Networks Model . 74 4.5.2 Simulation of CRNs . 76 4.5.3 Grid gate . 78 4.5.4 Modifications to grid gate for simulating CRNs . 79 5 Verification Software 89 5.1 Preliminaries . 89 5.1.1 Definitions . 89 5.1.2 Solvers . 91 5.2 Computing stable configurations of TBNs . 92 5.2.1 Finding stable configurations of TBNs . 92 5.2.2 Casting StableConfigs as an IP . 93 iv 5.2.3 Empirical running time measurements . 98 5.3 Computing bases of locally stable configurations of TBNs . 99 5.3.1 Equivalence of polymer bases and Hilbert bases . 99 5.3.2 Using the polymer basis to reason about TBN behavior . 102 5.3.3 A case example: Circular Translator Cascade . 103 6 Grid Gate Experiment 106 6.1 Designs . 106 6.2 Designing Orthogonal Domains . 110 6.2.1 Constraints . 110 6.3 Results . 116 6.3.1 Assay technique . 116 6.3.2 Stable formation of Grid Gate . 118 6.4 Next Steps . 123 v Abstract Design and Verification of Modular Components in Thermodynamic Binding Networks Designing engineered molecular systems typically requires specialized knowledge of the particular substrate; however, one can also reason about such systems in a substrate- independent fashion, by examining the underlying energetics that govern any chemical substrate: the formation of molecular bonds and the number of complexes formed. The thermodynamic binding networks (TBN) model was developed to study such systems, and in particular, to determine fault-tolerance in molecular systems such as DNA strand displacement cascades. This dissertation details an extended form of the model in which complexes can merge together or split apart at an energetic benefit/cost. This exten- sion allows one to also reason about reachability of configurations with respect to energy barriers. Several theoretical constructions are presented here which demonstrate that such energy barriers can be programmably large, implement catalytic and autocatalytic behavior, and be part of larger, modular systems in which complex behavior can be real- ized. Indeed, reasoning about the energy barrier between configurations in such systems is proved here to be PSPACE-hard, even to a c-factor approximation. This dissertation also contains details of integer and constraint programming formulations that can solve certain questions related to a system's energetics. Also made formal here is the con- nection between TBNs and the well-studied combinatorial concept of Hilbert bases, and examples are given which illustrate how one can use a Hilbert basis to verify particular aspects of TBN designs. Finally, the details of an experiment attempting to implement one of the programmable TBN constructions are given, along with empirical results and interpretations. vi Contents 1 Chapter 1 Introduction Recent experimental breakthroughs in DNA nanotechnology [19] have enabled the con- struction of intricate molecular machinery whose complexity rivals that of other biological macromolecules, even executing general-purpose algorithms [59]. A major challenge in creating synthetic DNA molecules that undergo desired chemical reactions is the occur- rence of erroneous \leak" reactions [45], driven by the fact that the products of the leak reactions are more energetically favorable. A promising design principle to mitigate such errors is to build \thermodynamic robustness" into the system, ensuring that leak reac- tions incur an energetic cost [55, 57] by logically forcing one of two unfavorable events: either many molecular bonds must break|an \enthalpic" cost|or many separate molecu- lar complexes (called polymers in this document) must simultaneously come together|an \entropic" cost. The model of thermodynamic binding networks (TBNs) [25] was defined as a combina- torial abstraction of such molecules, deliberately simplifying substrate-dependent details of DNA in order to isolate the foundational energetic contributions of forming bonds and separating polymers. A TBN consists of monomers containing specific binding sites, where binding site a can bind only to its complement a∗. A key aspect of the TBN model is the lack of geometry: a monomer is an unordered collection of binding sites such as fa; a; b∗; cg.A configuration of a TBN describes which monomers are grouped into poly- mers; bonds can only form within a polymer. One can formalize the \correctness" of a TBN by requiring that its desired configuration(s) be stable: the configuration maximizes 2 the number of bonds formed, a.k.a., it is saturated, and, among all saturated configura- tions, it maximizes the number of separate polymers.1 See Fig. 1.1 for an example. Stable configurations are meant to capture the minimum free energy structures of the TBN. Un- fortunately, answering basic questions such as \Is a particular TBN configuration stable?" turn out to be NP-hard [11]. a b a b a b a b a b a b a b a* b* a* b* a* b* a* b* a* b* a* b* a* b* a b a b a b a b a b a b a b not saturated saturated stable Figure 1.1: Example of a simple thermodynamic binding network (TBN). There are four monomers: ∗ ∗ m1 = fa ; b g; m2 = fa; bg; m3 = fag; m4 = fbg; with seven configurations shown: four of these configu- rations are saturated because they have the maximum of 2 bonds. Of these, three have 2 polymers and one has 3 polymers, making the latter the only stable configuration. Despite the suggestive lines between binding sites, the model of this paper ignores individual bonds, defining a configuration solely by how it partitions the set of monomers into polymers, assuming that a maximum number of bonds will form within each polymer. (Thus other configurations exist besides those shown, which would merge polymers shown without allowing new bonds to form.) Abstract mathematical models of molecular systems, such as chemical reaction net- works, have long been useful in natural science to study the properties of natural molecules. For a chemical system designed to perform computation, we can prescribe a chemical pro- gram with abstract chemical reactions such as A + C ! B + C (1.1) A ! B: (1.2) In particular, a program may require Eq. 1.1 and forbid Eq. 1.2. But what remains hidden at this level of abstraction is a well-known chemical constraint: if Eq. 1.1 is possible, then Eq. 1.2 must also be, no matter the exact substances. Knowing this, we might try to slow Eq. 1.2 by ensuring B has high free energy. But then B + C must also have high free energy, so Eq. 1.1 slows in tandem. The only option to slow Eq. 1.2 but not Eq. 1.1 is to 1This definition captures the limiting case (often approximated in practice in DNA nanotechnology) corresponding to increasing the strength of bonds, while diluting (increasing volume), such that the ratio of binding to unbinding rate goes to infinity. 3 use a kinetic barrier: designing A so that, although it is possible for A to reconfigure into B, the system must traverse a higher energy (less favorable) intermediate in the absence of C. It seems difficult to engineer kinetic energy barriers and catalysis in a way that is independent of the particular chemical substrate.
Load more

Recommended publications
  • Graeco-Latin Squares and a Mistaken Conjecture of Euler Author(S): Dominic Klyve and Lee Stemkoski Source: the College Mathematics Journal, Vol
    Graeco-Latin Squares and a Mistaken Conjecture of Euler Author(s): Dominic Klyve and Lee Stemkoski Source: The College Mathematics Journal, Vol. 37, No. 1 (Jan., 2006), pp. 2-15 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/27646265 . Accessed: 15/10/2013 14:08 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The College Mathematics Journal. http://www.jstor.org This content downloaded from 128.195.64.2 on Tue, 15 Oct 2013 14:08:26 PM All use subject to JSTOR Terms and Conditions Graeco-Latin Squares and a Mistaken Conjecture of Euler Dominic Klyve and Lee Stemkoski Dominic Klyve ([email protected]) and Lee Stemkoski ([email protected]) are both Ph.D. students inmathematics at Dartmouth College (Hanover, NH 03755). While their main area of study is number theory (computational and analytic, respectively), their primary avocation is the history of mathematics, in particular the contributions of Leonhard Euler. They are the creators and co-directors of the Euler Archive (http://www.eulerarchive.org), an online resource which is republishing the works of Euler in their original formats.
    [Show full text]
  • 2018 Joint Meetings of the Florida Section of the Mathematical Association of America and the Florida Two-Year College Mathematics Association
    2018 Joint Meetings Of The Florida Section Of The Mathematical Association of America And The Florida Two-Year College Mathematics Association Florida Atlantic University Davie Campus February 9 - 10, 2018 Florida Section of the Mathematical Association of America 2017 – 2018 Section Representative Pam Crawford, Jacksonville University President Brian Camp, Saint Leo University Past President John R. Waters Jr., State College of Florida Vice-President for Programs Charles Lindsey, Florida Gulf Coast University Vice-President for Site Selection Anthony Okafor, University of West Florida Secretary-Treasurer David Kerr, Eckerd College Newsletter Editor Daniela Genova, University of North Florida Coordinator of Student Activities Donald Ransford, Florida SouthWestern State College Webmaster Altay Özgener, State College of Florida President-elect Penny Morris, Polk State College VP for Programs-elect Milé Krajčevski, University of South Florida VP for Site Selection-elect Joy D’Andrea, USF Sarasota-Manatee Florida Two-Year College Mathematics Association 2017-2018 President Altay Özgener, State College of Florida Past President Ryan Kasha, Valencia College Vice-President for Programs Don Ransford, Florida SouthWestern State College Secretary Sidra Van De Car, Valencia College Treasurer Ryan Kasha, Valencia College Newsletter Editor Rebecca Williams, State College of Florida Membership Chair Dennis Runde, State College of Florida Historian Joni Pirnot, State College of Florida Webmaster Altay Özgener, State College of Florida President-elect Sandra Seifert, Florida SouthWestern State College 2 PROGRAM Friday, February 9, 2018 Committee Meetings and Workshops FL – MAA 9:30 – 11:00 Executive Committee Meeting LA 150 FTYCMA 9:30 – 10:30 FTYCMA Officer’s Meeting LA 132 10:30 – 12:00 FTYCMA Annual Business Meeting LA 132 12:00 – 1:30 New Members Luncheon and Mingle Student Union (BC54) Free lunch for first time attendees.
    [Show full text]
  • "Read Euler, Read Euler, He Is the Master of Us All."
    "Read Euler, read Euler, he is the master of us all." © 1997−2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non−commercial use. For other uses, including electronic redistribution, please contact us. March 2007 Features "Read Euler, read Euler, he is the master of us all." by Robin Wilson Leonhard Euler was the most prolific mathematician of all time. He wrote more than 500 books and papers during his lifetime about 800 pages per year with an incredible 400 further publications appearing posthumously. His collected works and correspondence are still not completely published: they already fill over seventy large volumes, comprising tens of thousands of pages. Euler worked in an astonishing variety of areas, ranging from the very pure the theory of numbers, the geometry of a circle and musical harmony via such areas as infinite series, logarithms, the calculus and mechanics, to the practical optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides. Indeed, Euler originated so many ideas that his successors have been kept busy trying to follow them up ever since; indeed, to Pierre−Simon Laplace, the French applied mathematician, is attributed the exhortation in the title of this article. "Read Euler, read Euler, he is the master of us all." 1 "Read Euler, read Euler, he is the master of us all." Leonhard Euler 1707 − 1783. He would have turned 300 this year. Not surprisingly, many concepts are named after him: Euler's constant, Euler's polyhedron formula, the Euler line of a triangle, Euler's equations of motion, Eulerian graphs, Euler's pentagonal formula for partitions, and many others.
    [Show full text]
  • Carver Award: Lynne Billard We Are Pleased to Announce That the IMS Carver Medal Committee Has Selected Lynne CONTENTS Billard to Receive the 2020 Carver Award
    Volume 49 • Issue 4 IMS Bulletin June/July 2020 Carver Award: Lynne Billard We are pleased to announce that the IMS Carver Medal Committee has selected Lynne CONTENTS Billard to receive the 2020 Carver Award. Lynne was chosen for her outstanding service 1 Carver Award: Lynne Billard to IMS on a number of key committees, including publications, nominations, and fellows; for her extraordinary leadership as Program Secretary (1987–90), culminating 2 Members’ news: Gérard Ben Arous, Yoav Benjamini, Ofer in the forging of a partnership with the Bernoulli Society that includes co-hosting the Zeitouni, Sallie Ann Keller, biannual World Statistical Congress; and for her advocacy of the inclusion of women Dennis Lin, Tom Liggett, and young researchers on the scientific programs of IMS-sponsored meetings.” Kavita Ramanan, Ruth Williams, Lynne Billard is University Professor in the Department of Statistics at the Thomas Lee, Kathryn Roeder, University of Georgia, Athens, USA. Jiming Jiang, Adrian Smith Lynne Billard was born in 3 Nominate for International Toowoomba, Australia. She earned both Prize in Statistics her BSc (1966) and PhD (1969) from the 4 Recent papers: AIHP, University of New South Wales, Australia. Observational Studies She is probably best known for her ground-breaking research in the areas of 5 News from Statistical Science HIV/AIDS and Symbolic Data Analysis. 6 Radu’s Rides: A Lesson in Her research interests include epidemic Humility theory, stochastic processes, sequential 7 Who’s working on COVID-19? analysis, time series analysis and symbolic 9 Nominate an IMS Special data. She has written extensively in all Lecturer for 2022/2023 these areas, publishing over 250 papers in Lynne Billard leading international journals, plus eight 10 Obituaries: Richard (Dick) Dudley, S.S.
    [Show full text]
  • An Introduction to SDR's and Latin Squares
    Morehead Electronic Journal of Applicable Mathematics Issue 4 — MATH-2005-03 Copyright c 2005 An introduction to SDR’s and Latin squares1 Jordan Bell2 School of Mathematics and Statistics Carleton University, Ottawa, Ontario, Canada Abstract In this paper we study systems of distinct representatives (SDR’s) and Latin squares, considering SDR’s especially in their application to constructing Latin squares. We give proofs of several important elemen- tary results for SDR’s and Latin squares, in particular Hall’s marriage theorem and lower bounds for the number of Latin squares of each order, and state several other results, such as necessary and sufficient conditions for having a common SDR for two families. We consider some of the ap- plications of Latin squares both in pure mathematics, for instance as the multiplication table for quasigroups, and in applications, such as analyz- ing crops for differences in fertility and susceptibility to insect attack. We also present a brief history of the study of Latin squares and SDR’s. 1 Introduction and history We first give a definition of Latin squares: Definition 1. A Latin square is a n × n array with n distinct entries such that each entry appears exactly once in each row and column. Clearly at least one Latin square exists of all orders n ≥ 1, as it could be made trivially with cyclic permutations of (1, . , n). This is called a circulant matrix of (1, . , n), seen in Figure 1. We discuss nontrivial methods for generating more Latin squares in Section 3. Euler studied Latin squares in his “36 officers problem” in [7], which had six ranks of officers from six different regiments, and which asked whether it is possible that no row or column duplicate a rank or a regiment; Euler was unable to produce such a square and conjectured that it is impossible for n = 4k + 2, and indeed the original case k = 1 of this claim was proved by G.
    [Show full text]
  • The Problem of the 36 Officers Kalei Titcomb
    The Problem of the 36 Officers Kalei Titcomb June 12, 2009 Contents 1 Latin Squares 5 2 Orthogonal Latin Squares 6 3 Designs 7 3.1 Transversal Designs (TDs) . 7 3.2 Pairwise balanced Designs (PBDs) . 8 4 The incidence matrix M of a PBD 9 4.1 Incidence matrix for 4 × 4 case . 11 4.2 Incidence matrix M for the 6 × 6 case (assuming such a case exists) . 12 5 The code C of P and Lemma I 13 6 Even sub-PBD and Lemma II 14 7 The mysterious case of m = 8 and Lemma III 16 7.1 The new incidence matrix: Mb ............................. 18 7.2 A graph named G ................................... 19 8 Property A and Lemma IV 21 9 The last hope and Lemma V 28 10 Curriculum Activity I 29 10.1 Latin Squares . 29 10.2 Latin Squares–key . 31 10.3 Reflecting on Activity I . 32 11 Curriculum Activity II 33 11.1 Orthogonal Latin Squares . 33 11.2 Orthogonal Latin Squares–key . 34 11.3 Reflecting on Activity II . 35 12 Curriculum Activity III 36 12.1 Magic Squares . 36 12.2 Magic Squares–key . 37 12.3 Reflecting on Activity III . 39 1 13 Curriculum Activity IV 40 13.1 Forming magic squares . 40 13.2 Finding Magic squares–key . 42 13.3 Reflecting on Activity IV . 44 2 Introduction to the Mathematical Exploration Ever since I was a little kid, I have enjoyed doing puzzles. While my parents bought the crossword puzzle books, I always wanted the logic books because they had games, and a variety of puzzles including crossword.
    [Show full text]
  • Professor Stewart's Casebook of Mathematical Mysteries
    Professor Stewart’s Casebook of Mathematical Mysteries Professor Ian Stewart is known throughout the world for making mathematics popular. He received the Royal Society’s Faraday Medal for furthering the public understanding of science in1995, the IMA Gold Medal in 2000, the AAAS Public Understanding of Science and Technology Award in 2001 and the LMS/IMA Zeeman Medal in 2008. He was elected a Fellow of the Royal Society in 2001. He is Emeritus Professor of Mathematics at the University of Warwick, where he divides his time between research into nonlinear dynamics and furthering public awareness of mathematics. His many popular science books include (with Terry Pratchett and Jack Cohen) The Science of Discworld I to IV, The Mathematics of Life, 17 Equations that Changed the World and The Great Mathematical Problems. His app, Professor Stewart’s Incredible Numbers, was published jointly by Profile and Touch Press in March 2014. By the Same Author Concepts Of Modern Mathematics Game, Set, And Math Does God Play Dice? Another Fine Math You’ve Got Me Into Fearful Symmetry Nature’s Numbers From Here To Infinity The Magical Maze Life’s Other Secret Flatterland What Shape Is A Snowflake? The Annotated Flatland Math Hysteria The Mayor Of Uglyville’s Dilemma How To Cut A Cake Letters To A Young Mathematician Taming The Infinite (Alternative Title: The Story Of Mathematics) Why Beauty Is Truth Cows In The Maze Mathematics Of Life Professor Stewart’s Cabinet Of Mathematical Curiosities Professor Stewart’s Hoard Of Mathematical Treasures Seventeen
    [Show full text]
  • Coloring Cayley Tables of Finite Groups
    Coloring Cayley Tables of Finite Groups by Kevin C. Halasz B.Sc., University of Puget Sound, 2014 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Faculty of Science c Kevin C. Halasz 2017 SIMON FRASER UNIVERSITY Summer 2017 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Kevin C. Halasz Degree: Master of Science (Mathematics) Title: Coloring Cayley Tables of Finite Groups Examining Committee: Chair: Ralf Wittenberg Associate Professor Luis Goddyn Senior Supervisor Professor Matt DeVos Supervisor Associate Professor Ladislav Stacho Internal Examiner Associate Professor Date Defended: August 8, 2017 ii Abstract The chromatic number of a latin square L, denoted χ(L), is defined as the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square L satisfies χ(L) ≤ |L| + 2. If true, this would resolve a longstanding conjecture, commonly attributed to Brualdi, that every latin square has a partial transversal of length |L|−1. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we constructively show that all finite Abelian groups G have Cayley tables with chromatic number |G| + 2. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For |G| ≥ 3, this improves the best-known general upper bound 3 from 2|G| to 2 |G|, while yielding an even stronger result in infinitely many cases.
    [Show full text]
  • Graeco-Latin Squares and a Mistaken Conjecture of Euler∗
    Graeco-Latin Squares and a Mistaken Conjecture of Euler∗ Dominic Klyve Lee Stemkoski Introduction Late in his long and productive career, Leonhard Euler published a hundred-page paper detailing the properties of a new mathematical structure: Graeco-Latin squares. In this paper, Euler claimed that a Graeco-Latin square of size n could never exist for any n of the form 4k +2, although he was not able to prove it. In the end, his difficulty was validated. Over a period of 200 years, more than twenty researchers from five countries worked on the problem. Even then, they succeeded only after using techniques from many branches of mathematics including group theory, finite fields, projective geometry, and statistical and block designs; eventually, modern computers were employed to finish the job. A Latin square (of order n) is an n-by-n array of n distinct symbols (usually the integers 1, 2, . , n) in which each symbol appears exactly once in each row and column. Some examples appear in Figure 1. 1 2 4 3 5 4 3 1 2 1 2 3 4 5 2 1 3 3 4 2 1 3 1 2 3 4 1 5 2 1 2 4 3 2 3 1 2 3 5 4 1 2 1 3 4 5 1 3 2 4 Figure 1: Latin squares of orders 3, 4, and 5 A Graeco-Latin square (of order n) is an n-by-n array of ordered pairs from a set of n symbols such that in each row and each column of the array, each symbol appears exactly once in each coordinate, and that each of the n2 possible pairs appears exactly once in the entire square.
    [Show full text]
  • On Orthogonal Latin Squares
    On Orthogonal Latin Squares Yanling Chen NTNU Crypto-Seminar, Nov 9-10, Bergen www.ntnu.no Yanling Chen, On Orthogonal Latin Squares Examples 1234 123 12 2341 1 231 21 3412 ··· 312 4123 2 Latin Square (LS) Definition A Latin square of order n is an n-by-n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. www.ntnu.no Yanling Chen, On Orthogonal Latin Squares 2 Latin Square (LS) Definition A Latin square of order n is an n-by-n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. Examples 1234 123 12 2341 1 231 21 3412 ··· 312 4123 www.ntnu.no Yanling Chen, On Orthogonal Latin Squares Example 123 1 2 3 11 22 33 231 3 1 2 23 31 12 ⊥ ⇒ 312 2 3 1 32 13 21 3 Orthogonal Latin Square Definition 2 Latin squares of the same order n are said to be orthogonal if when they overlap, each of the possible n2 ordered pairs occur exactly once. www.ntnu.no Yanling Chen, On Orthogonal Latin Squares 3 Orthogonal Latin Square Definition 2 Latin squares of the same order n are said to be orthogonal if when they overlap, each of the possible n2 ordered pairs occur exactly once. Example 123 1 2 3 11 22 33 231 3 1 2 23 31 12 ⊥ ⇒ 312 2 3 1 32 13 21 www.ntnu.no Yanling Chen, On Orthogonal Latin Squares 4 History Leonhard Euler [Euler 1782] the problem of 36 officers, 6 ranks, 6 regiments • he concluded that no two 6 6LSareorthogonal • × L.
    [Show full text]
  • A Distributed System for Enumerating Main Classes of Sets of Mutually Orthogonal Latin Squares
    A distributed system for enumerating main classes of sets of mutually orthogonal Latin squares Johannes Gerhardus Benad´e Thesis presented in partial fulfilment of the requirements for the degree of Master of Science in the Faculty of Science at Stellenbosch University Supervisor: Prof JH van Vuuren Co-supervisor: Dr AP Burger December 2014 Declaration By submitting this thesis electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly oth- erwise stated), that reproduction and publication thereof by Stellenbosch University will not infringe any third party rights and that I have not previously in its entirety or in part submitted it for obtaining any qualification. Date: December 1, 2014 Copyright c 2014 Stellenbosch University All rights reserved i ii Abstract A Latin square is an n × n array containing n copies of each of n distinct symbols in such a way that no symbol is repeated in any row or column. Two Latin squares are orthogonal if, when superimposed, the ordered pairs in the n2 cells are all distinct. This notion of orthogonality ex- tends naturally to sets of k > 2 mutually orthogonal Latin squares (abbreviated in the literature as k-MOLS), which find application in scheduling problems and coding theory. In these instances it is important to differentiate between structurally different k-MOLS. It is thus useful to classify Latin squares and k-MOLS into equivalence classes according to their structural properties | this thesis is concerned specifically with main classes of k-MOLS, one of the largest equivalence classes of sets of Latin squares.
    [Show full text]
  • From Magic Squares to Sudoku
    Anything but square: from magic squares to Sudoku © 1997−2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non−commercial use. For other uses, including electronic redistribution, please contact us. March 2006 Features Anything but square: from magic squares to Sudoku by Hardeep Aiden What is a magic square? There is an ancient Chinese legend that goes something like this. Some three thousand years ago, a great flood happened in China. In order to calm the vexed river god, the people made an offering to the river Lo, but he could not be appeased. Each time they made an offering, a turtle would appear from the river. One day a boy noticed marks on the back of the turtle that seemed to represent the numbers 1 to 9. The numbers were arranged in such a way that each line added up to 15. Hence the people understood that their offering was not the right amount. Anything but square: from magic squares to Sudoku 1 Anything but square: from magic squares to Sudoku Nymph of the Lo River, an ink drawing on a handscroll, Ming dynasty, 16th century. Freer Gallery of Art The markings on the back of the turtle were in fact a magic square. A magic square is a square grid filled with numbers, in such a way that each row, each column, and the two diagonals add up to the same number. Here's what the magic square from the Lo Shu would have looked like. It has three rows and three columns, and if you add up the numbers in any row, column or diagonal, you always get 15.
    [Show full text]