synchronization and permutation groups
play

Synchronization and permutation groups Peter J. Cameron - PowerPoint PPT Presentation

Dec 26, 2023 •350 likes •1.27k views

Synchronization and permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk 4-ICC, Auckland December 2008 This is part of an investigation involving, among others, Jo ao Ara ujo, eter Neumann, Jan Saxl, Csaba Schneider, Pablo

  1. Synchronization and permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk 4-ICC, Auckland December 2008

  2. This is part of an investigation involving, among others, Jo˜ ao Ara´ ujo, Π eter Neumann, Jan Saxl, Csaba Schneider, Pablo Spiga, and Ben Steinberg. Cristy Kazanidis, Nik Ruskuc, Colva Roney-Dougal, Ian Gent and Tom Kelsey have also been involved.

  3. This is part of an investigation involving, among others, Jo˜ ao Ara´ ujo, Π eter Neumann, Jan Saxl, Csaba Schneider, Pablo Spiga, and Ben Steinberg. Cristy Kazanidis, Nik Ruskuc, Colva Roney-Dougal, Ian Gent and Tom Kelsey have also been involved. There is far more material than can be presented here; I will talk about other aspects of this topic in Perth next month. See you there!

  4. This is part of an investigation involving, among others, Jo˜ ao Ara´ ujo, Π eter Neumann, Jan Saxl, Csaba Schneider, Pablo Spiga, and Ben Steinberg. Cristy Kazanidis, Nik Ruskuc, Colva Roney-Dougal, Ian Gent and Tom Kelsey have also been involved. There is far more material than can be presented here; I will talk about other aspects of this topic in Perth next month. See you there! See also Gordon Royle’s talk at this meeting for a more combinatorial approach.

  5. Automata An automaton is a machine which can be in any of a set of internal states which cannot be directly observed.

  6. Automata An automaton is a machine which can be in any of a set of internal states which cannot be directly observed. We can force the machine to make any desired sequence of transitions (each transition being a mapping from the set of states to itself).

  7. Automata An automaton is a machine which can be in any of a set of internal states which cannot be directly observed. We can force the machine to make any desired sequence of transitions (each transition being a mapping from the set of states to itself). We can represent an automaton as an edge-coloured directed graph, where the vertices are the states, and the colours are the transitions. We require that the graph should have exactly one edge of each colour leaving each vertex.

  8. Synchronization Suppose that you are given an automaton (whose structure you know) in an unknown state. You would like to put it into a known state, by applying a sequence of transitions to it. Of course this is not always possible!

  9. Synchronization Suppose that you are given an automaton (whose structure you know) in an unknown state. You would like to put it into a known state, by applying a sequence of transitions to it. Of course this is not always possible! A reset word is a sequence of transitions which take the automaton from any state into a known state; in other words, the composition of the corresponding transitions is a constant mapping.

  10. An example ← 1 2 . . . . . . . ✉ . . . . . . ✉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . տ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ց . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . ց . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ↑ ր � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✉ . . . . . . . ✉ 4 3 ←

  11. An example ← 1 2 . . . . . . . ✉ . . . . . . ✉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . տ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ց . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . ց . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ↑ ր � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✉ . . . . . . . ✉ 4 3 ← You can check that (Blue, Red, Blue, Blue) is a reset word which takes you to room 3 no matter where you start.

  12. Applications ◮ Industrial robotics: pieces arrive to be assembled by a robot. The orientation is critical. You could equip the robot with vision sensors and manipulators so that it can rotate the pieces into the correct orientation. But it is much cheaper and less error-prone to regard the possible orientations of the pieces as states of an automaton on which transitions can be performed by simple machinery, and apply a reset word before the pieces arrive at the robot.

Recommend

The diameter of permutation groups permutation groups H. A. Helfgott February 2017 The

The diameter of permutation groups permutation groups H. A. Helfgott February 2017 The

The diameter of permutation groups H. A. Helfgott Introduction Diameter bounds New work on The diameter of permutation groups permutation groups H. A. Helfgott February 2017 The diameter of Cayley graphs permutation groups H. A.

751 views • 64 slides
Growth in permutation groups and linear New work on algebraic groups permutation groups H. A.

Growth in permutation groups and linear New work on algebraic groups permutation groups H. A.

Growth in permutation groups and linear algebraic groups H. A. Helfgott Introduction Diameter bounds Growth in permutation groups and linear New work on algebraic groups permutation groups H. A. Helfgott September 2018 Growth in

697 views • 65 slides
J , the all- 1 matrix. (The corresponding only G -invariant association scheme is the trivial A

J , the all- 1 matrix. (The corresponding only G -invariant association scheme is the trivial A

Association schemes and Permutation groups permutation groups Let G be a permutation group on . Then the characteristic functions of the orbits of G on Peter J Cameron G

44 views • 3 slides
Permutation Groups John Bamberg, Michael Giudici and Cheryl Praeger Centre for the Mathematics of

Permutation Groups John Bamberg, Michael Giudici and Cheryl Praeger Centre for the Mathematics of

Notation Basics of permutation group theory Arc-transitive graphs Primitivity Normal subgroups of primitive groups Graphs from groups Permutation Groups John Bamberg, Michael Giudici and Cheryl Praeger Centre for the Mathematics of Symmetry

579 views • 25 slides
Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk Groups and their Applications Manchester, 14 February 2007 In the beginning . . . A derangement is a permutation with no fixed points. In the

1.14k views • 77 slides
Permutation Groups and Transformation Semigroups Lecture 1: Introduction Peter J. Cameron

Permutation Groups and Transformation Semigroups Lecture 1: Introduction Peter J. Cameron

Permutation Groups and Transformation Semigroups Lecture 1: Introduction Peter J. Cameron University of St Andrews Shanghai Jiao Tong University 14 November 2017 Permutation groups For any set , Sym ( ) denotes the symmetric group of

2.01k views • 91 slides
Invariance groups of finite functions and orbit equivalence of permutation groups Tam as

Invariance groups of finite functions and orbit equivalence of permutation groups Tam as

Invariance groups of finite functions and orbit equivalence of permutation groups Tam as Waldhauser University of Szeged NSAC 2013 Novi Sad, 7th June 2013 Joint work with Eszter Horv ath, Reinhard P oschel, G eza Makay.

732 views • 72 slides
Primitive permutation groups with finite stabilizers Simon M. Smith City Tech, CUNY and The

Primitive permutation groups with finite stabilizers Simon M. Smith City Tech, CUNY and The

Primitive permutation groups with finite stabilizers Simon M. Smith City Tech, CUNY and The University of Western Australia Groups St Andrews 2013, St Andrews Primitive permutation groups A transitive group G of permutations of a set is

631 views • 27 slides
The diameter of permutation groups Proof ideas H. A. Helfgott and . Seress July 2013 Cayley

The diameter of permutation groups Proof ideas H. A. Helfgott and . Seress July 2013 Cayley

The diameter of permutation groups H. A. Helfgott and . Seress Introduction Alternating groups The diameter of permutation groups Proof ideas H. A. Helfgott and . Seress July 2013 Cayley graphs The diameter of permutation groups

586 views • 54 slides
Algorithms for Permutation groups Alice Niemeyer UWA, RWTH Aachen Alice Niemeyer (UWA, RWTH

Algorithms for Permutation groups Alice Niemeyer UWA, RWTH Aachen Alice Niemeyer (UWA, RWTH

Algorithms for Permutation groups Alice Niemeyer UWA, RWTH Aachen Alice Niemeyer (UWA, RWTH Aachen) Perm Groups Sommerschule 2011 1 / 36 Permutation Groups Permutation Groups The Symmetric Group Let be a finite set. The Symmetric group,

584 views • 37 slides
Oligomorphic permutation groups: growth rates and algebras Peter J. Cameron

Oligomorphic permutation groups: growth rates and algebras Peter J. Cameron

Oligomorphic permutation groups: growth rates and algebras Peter J. Cameron p.j.cameron@qmul.ac.uk Gregynog Mathematics Colloquium22 May 2007 The definition Examples, 2 Consider the group S r acting on the disjoint union Let G be a permutation

304 views • 4 slides
The diameter of permutation groups kos Seress May 2012 Cayley graphs The diameter of

The diameter of permutation groups kos Seress May 2012 Cayley graphs The diameter of

The diameter of permutation groups kos Seress The diameter of permutation groups kos Seress May 2012 Cayley graphs The diameter of permutation groups kos Seress Definition G = S is a group. The Cayley graph ( G , S ) has

531 views • 19 slides
Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk Groups and their ApplicationsManchester, 14 February 2007 uous closed curve in S which cannot be lifted In the beginning . . . A derangement is a

231 views • 5 slides
Primitive permutation groups and generalised quadrangles Tomasz Popiel (QMUL & UWA) Joint

Primitive permutation groups and generalised quadrangles Tomasz Popiel (QMUL & UWA) Joint

Primitive permutation groups and generalised quadrangles Tomasz Popiel (QMUL & UWA) Joint work with John Bamberg and Cheryl E. Praeger Groups St. Andrews in Birmingham 8 August 2017 Tomasz Popiel (QMUL & UWA) Primitive groups and GQs

205 views • 9 slides
Basic Algorithms for Permutation Groups Alexander Hulpke Department of Mathematics Colorado

Basic Algorithms for Permutation Groups Alexander Hulpke Department of Mathematics Colorado

Basic Algorithms for Permutation Groups Alexander Hulpke Department of Mathematics Colorado State University Fort Collins, CO, 80523-1874 Arizona Summer Program 2008 JAH, Arizona Summer Program 2008 Basic Algorithms for Permutation Groups 1

386 views • 22 slides
Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial

Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial

Orbit coherence in permutation groups John R. Britnell Department of Mathematics Imperial College London j.britnell@imperial.ac.uk Groups St Andrews 2013 Joint work with Mark Wildon (RHUL) Orbit partitions Let G be a group of permutations of

565 views • 33 slides
Quantum Increasing Sequences generate Quantum Permutation Groups Pawe l J oziak Faculty

Quantum Increasing Sequences generate Quantum Permutation Groups Pawe l J oziak Faculty

Quantum Increasing Sequences generate Quantum Permutation Groups Pawe l J oziak Faculty of Mathematics and Information Technology, Warsaw University of Technology 8 VIII 2019 Universitetet i Oslo Quantum groups and their analysis I

686 views • 67 slides
Permutation groups and transformation semigroups Peter J. Cameron University of St Andrews

Permutation groups and transformation semigroups Peter J. Cameron University of St Andrews

Permutation groups and transformation semigroups Peter J. Cameron University of St Andrews Groups St Andrews, August 2013 Groups and semigroups How can group theory help the study of semigroups? Groups and semigroups How can group theory

1.22k views • 96 slides
Permutation Groups and Transformation Semigroups Lecture 3: Regularity Peter J. Cameron

Permutation Groups and Transformation Semigroups Lecture 3: Regularity Peter J. Cameron

Permutation Groups and Transformation Semigroups Lecture 3: Regularity Peter J. Cameron University of St Andrews Shanghai Jiao Tong University November 2017 The prototype result Recall the theoreom of Ara ujo, Mitchell and Schneider in

1.17k views • 84 slides
Subdegrees of primitive permutation groups Michael Giudici Centre for the Mathematics of Symmetry

Subdegrees of primitive permutation groups Michael Giudici Centre for the Mathematics of Symmetry

Subdegrees of primitive permutation groups Michael Giudici Centre for the Mathematics of Symmetry and Computation G2D2 2019, Yichang, China Subdegrees G Sym() transitive on a finite set . Let and G be the stabiliser of

492 views • 23 slides
Transitive permutation groups: Minimal, invariable and random generation Gareth Tracey

Transitive permutation groups: Minimal, invariable and random generation Gareth Tracey

Transitive permutation groups: Minimal, invariable and random generation Gareth Tracey University of Warwick Bielefeld, January 12th, 2017 A motivational question How many subgroups does the symmetric group S n have? A motivational question

1.4k views • 98 slides
Permutation groups and transformation semigroups Peter J. Cameron Novi Sad Algebraic Conference,

Permutation groups and transformation semigroups Peter J. Cameron Novi Sad Algebraic Conference,

Permutation groups and transformation semigroups Peter J. Cameron Novi Sad Algebraic Conference, June 2013 Groups and semigroups How can group theory help the study of semigroups? Groups and semigroups How can group theory help the study of

923 views • 91 slides
Semi-Latin squares, permutation groups, and statistical optimality Leonard Soicher Queen Mary,

Semi-Latin squares, permutation groups, and statistical optimality Leonard Soicher Queen Mary,

Semi-Latin squares, permutation groups, and statistical optimality Leonard Soicher Queen Mary, University of London Ottawa-Carleton Discrete Mathematics Days: 13-14 May 2011 Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality

745 views • 62 slides
Operator algebraic properties for free wreath products by quantum permutation groups -Conf

Operator algebraic properties for free wreath products by quantum permutation groups -Conf

Fusion rules for free wreath products Operator algebraic properties for free wreath products by quantum permutation groups -Conf erence du GDRE Noncommutative Geometry and Applications- Fran cois Lemeux (part. joint work with Pierre

326 views • 19 slides

More recommend