DOCSLIB.ORG

Association Schemes Association Schemes: Definition an Association

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Association Schemes Association Schemes: Definition an Association Association schemes: definition Association schemes An association scheme of rank r on a finite set Ω (sometimes called points) R. A. Bailey is a colouring of the elements of Ω Ω [email protected], [email protected] × (sometimes called edges) by r colours such that (i) one colour is exactly the main diagonal; G. C. Steward lecture, (ii) each colour is symmetric about the main diagonal; Gonville and Caius College, Cambridge (iii) if (α, β) is yellow yellow then there are exactly pred,blue points γ such that (α,γ) is red and (γ, β) is blue (for all values of yellow, red and blue). 10 May 2013 The non-diagonal classes are called associate classes. 1/42 2/42 An association scheme with 15 points and 3 associate An association scheme defined by a Latin square classes 1 2 3 4 If (α, β) is blue 1 A B C D α β 2 B C D A white blue 1 3 C D A B blue white 1 4 D A B C blue blue 1 black pink 4 pink black 4 pink pink 4 different cells in same row red different cells in same column green different cells in same letter yellow 3/42 other different cells black 4/42 The association scheme defined by the cube Association schemes: alternative definition ABCDEFGH EF A The adjacency matrix Ai for colour i is the Ω Ω matrix with @ × @ B 1 if (α, β) has colour i @ Ai(α, β) = @ C 0 otherwise. A B D Colour 0 is the diagonal, so E D C (i) A0 = I (identity matrix); @ F @ (ii) every Ai is symmetric; @ G (iii) A A = pk A ; @ i j ∑ ij k H k H G (iv) ∑Ai = J (all-1s matrix). edge (distance 1) yellow i distance 2 blue opposite (distance 3) red 5/42 6/42 One adjacency matrix for the cube That hard condition Ablue = ABCDEFGH AiAj(α, β) = ∑Ai(α,γ)Aj(γ, β) A 0 0 1 0 0 1 0 1 γ 0 0 0 1 1 0 1 0 = γ : (α,γ) has colour i and (γ, β) has colour j B |{ }| C 1 0 0 0 0 1 0 1 k = p if (α, β) has colour k. D 0 1 0 0 1 0 1 0 ij 0 1 0 1 0 0 1 0 E So k F 1 0 1 0 0 0 0 1 AiAj = ∑pijAk. k G 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 0 For an association scheme, H we turn counting into matrix multiplication. 7/42 8/42 How many edges of colour i at each point? How many edges of colour i at each point in this association scheme? AiAi(α,α) = ∑Ai(α,γ)Ai(γ,α) γ = γ : (α,γ) has colour i and (γ,α) has colour i |{ }| i ai = γ : (α,γ) has colour i by symmetry white 1 |{ }| purple 8 blue 4 0 = pii. black 2 0 Put ai = pii. Then there are precisely ai edges of colour i at each point. 9/42 10/42 The trivial association scheme on 10 points The group divisible association scheme GD(3,5). A0 = I The set Ω is A = J I 1 − partitioned into 3 “groups” of size 5. This is the only association scheme Different points in with rank 2. the same “group” are 1st associates (green). Points in different “groups” are 2nd associates (red). 11/42 12/42 Association schemes in design of experiments The Hamming association scheme Hamm(4,2). Suppose that we are going to do an experiment to compare 5 varieties of wheat. The points are We can use 15 plots of land in a single field. 4 vectors in Z2. If the plots are all alike, they form the trivial association scheme How many equal on 15 plots. coordinates? If the plots are not all alike, we may have to group them into 4 white three blocks: one block has the five plots near to the trees; 3 yellow another block has the five stony plots; and the third block has 2 black the five plots near to the stream. Now we have the group 1 green divisible association scheme GD(3,5). 0 red 13/42 14/42 The cube association scheme is Hamm(3,2) Association schemes in coding theory 000 001 011 010 100 101 111 110 You receive the following top-secret message: 100 101 000 @ SHOOT BORDS @ 001 @ @ 011 Should you 000 001 010 SHOOT BORIS? 100 011010 @ 101 SHOOT BIRDS? @ @ 111 SHOOT LORDS? @ 110 111110 SHOOT BONDS? edge (distance 1) yellow 2 equal coordinates distance 2 blue 1 equal coordinate Coded versions of possible messages should have opposite (distance 3) red 0 equal coordinates large Hamming distance between them. 15/42 16/42 The Johnson association scheme J(6,3). Association schemes in genetics 12 13 14 15 23 24 25 34 35 45 12 13 14 The points are 15 subsets of size 3 of a set of size 6. 23 What is the size of 24 their intersection? 25 34 3 white 2 green 35 1 black 45 0 red Types of offspring from cross-breeding five pure breeds of dog. Two different types may have one or zero parental lines in common. This is J(5,2). 17/42 18/42 How do we verify the hard condition? Where can I get to in two red steps? 12 13 14 15 23 24 25 34 35 45 We need to evaluate A A for 0 i r 1 and 0 j r 1. i j ≤ ≤ − ≤ ≤ − 12 A = I and IA = A I = A , so 0 i i i 13 we only need to evaluate AiAj for 1 i r 1 and 1 j r 1. ≤ ≤ − ≤ ≤ − 14 If Ai does not have a constant number of 1s in each row and column then we do not have an association scheme. 15 If Ai has exactly ai entries 1 in each row and column then 23 AiJ = aiJ. So 24 Ai(A0 + + Ar 1) = AiJ = aiJ = ai(A0 + + Ar 1). So ··· − ··· − 25 we only need to evaluate AiAj for 1 i r 2 and 1 j r 2. ≤ ≤ − ≤ ≤ − 34 If A A = ∑ pk A then A A is symmetric, so i j k ij k i j 35 k AiAj = (AiAj)> = A>A> = AjAi = ∑ p Ak, so j i k ij 45 we only need to evaluate A A for 1 i j r 2. i j ≤ ≤ ≤ − 2 A2 = 3I + 0A + 1A . When r = 3 we only need to evaluate A1. red red blue 19/42 20/42 Strongly regular graphs Association schemes of rank 3 A graph is strongly regular if 0 I every point is in p11 edges; 1 I every edge is in p11 triangles; 2 I every non-edge is in p11 two-paths; I the graph is neither null nor complete. ............... An association scheme has rank 3 ............. ............. ............. 12............. ............. ............. ............. ............. ............. ............. ............. ............. if and only if ............. .v ............. ............. ............. ............. ............. ............. ............. ............. ............. each of its non-identity classes forms the edges of a strongly ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. regular graph. ............. ...... ............. ............. ... ... 35 ............. ............. ... ... ............. ............. ... ... ............. ............. ... ... ............. ............ ... ... ............ .................... ... ... .................... ... .......................... ... v... .......................... ... ... ......................... ... ... ......................... ... 4534 ... .......................... ... ... .......................... ... ... .......................... ... ... .......................... ... 0 ... .................. ... ... .......................................................................................................................................................................................................................................................... ......... ... ... ............. ... ... ............. ... p = 3 v... ............. ... ... ............. ... v 11 ... ............. ... ... ............. ... ... ............. ... ... ............. ... ... 15 ............ ... ... ............ 23 ... ... ............. ... ... ............. ... ... v ............. ............. v ... ... ... ............. ............. ... ... ... ... ............. ............. ... ... 1 ... ... ............. ............. ... ... ... ... ..................... ... ... p = 0 ... ... .......................... ... ... 11 ... ... ............. ............. ... ... ... ... ............. ............. ... ... ... ... ............. ............. ... ... ... ... ............. ............. ... ... ... ................ ................ ... 2 ... ............ ............ ... ... ..... ..... ... p11 = 1 ... ..... .....2414 ... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... ... ..... v v ..... ... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... ........ ........ 25 . ....................................................................................................................................................................................................................................................................................................... 13 21/42 22/42 v v Distance-regular graphs The icosahedral association scheme Suppose that a graph G with v vertices is connected and simple (no multiple edges) and has diameter (maximum distance) d. Let A be the v v matrix whose (α, β)-entry is equal to i × 1 if the distance from α to β in G is i The 12 vertices and (0 otherwise 30 edges of the icosahedron form a distance-regular The graph G is distance-regular if A A is a linear combination 1 i graph with of Ai 1, Ai and Ai+1 for all i. − diameter 3. Theorem distance 1 turquoise If G is distance-regular with diameter d then its distance classes form distance 2 blue an association scheme of rank d + 1. distance 3 purple If r 4 then not all association schemes of rank r arise in this ≥ I have a wonderful cardboard model of this . but it was too way. large to fit on the train. 23/42 24/42 Cyclic association schemes (circuits) Association schemes and permutation groups The n-circuit is an association scheme with rank n 1 1 + − . 2 If G is a transitive permutation group on Ω, If the distance from α to β is d then (α, β) has colour d. it induces a permutation group on Ω Ω. × ........... ............. ............. Give (α, β) the same colour as (γ,δ) if and only if ............. ............. ............. ............. ............. ............
Load more

Recommended publications
  • Epidemiology, Statistics and Data Management
    THIRD JOINT CONFERENCE OF BHIVA AND BASHH 2014 Dr Carol Emerson The Royal Hospitals, Belfast 1-4 April 2014, Arena and Convention Centre Liverpool THIRD JOINT CONFERENCE OF BHIVA AND BASHH 2014 Dr Carol Emerson The Royal Hospitals, Belfast COMPETING INTEREST OF FINANCIAL VALUE > £1,000: Speaker Name Statement Dr Carol Emerson none declared Date April 2014 1-4 April 2014, Arena and Convention Centre Liverpool BASHH & BHIVA Mentoring Scheme for New Consultants & SAS doctors Dr Carol Emerson Consultant GU/HIV Medicine, Belfast Trust Overview What mentoring is The BASHH & BHIVA mentoring scheme Feedback from first wave participants Future plans for the mentoring scheme What is mentoring? SCOPME 1998 ‘A process whereby an experienced, highly regarded, empathic person (the mentor) guides another usually younger individual (the mentee) in the development and re-examination of their own ideas, learning and personal or professional development. The mentor, who often but not necessarily works in the same organisation or field as the mentee, achieves this by listening or talking in confidence to the mentee’ What is mentoring? Department of Health 2000 ....helping another person to become what that person aspires to be.... Mentoring for Doctors Supported by • Department of Health NHS Plan 2000 • British International Doctors Association – scheme since 1998 • BMA 2003 – lobbies for government funding to develop mentoring for all doctors • National Clinical Assessment Authority (NCAS) – encourages mentoring as a development / support tool / intervention for underperforming doctors • Academy of Medical Sciences • Royal Colleges • Royal College Psychiatrists • Royal College Paediatrics and Child • Royal College Obstetricians and Gynaecologists • Royal College Surgeons Mentoring and the GMC Good Medical Practice 2013 Domain 1: Knowledge skills and performances Develop and maintain your professional performance • 10 - ‘You should be willing to find and take part in structured support opportunities offered by your employer or contracting body (for example, mentoring).
    [Show full text]
  • Association Schemes1
    Association Schemes1 Chris Godsil Combinatorics & Optimization University of Waterloo ©2010 1June 3, 2010 ii Preface These notes provide an introduction to association schemes, along with some related algebra. Their form and content has benefited from discussions with Bill Martin and Ada Chan. iii iv Contents Preface iii 1 Schemes and Algebras1 1.1 Definitions and Examples........................1 1.2 Strongly Regular Graphs.........................3 1.3 The Bose-Mesner Algebra........................6 1.4 Idempotents................................7 1.5 Idempotents for Association Schemes.................9 2 Parameters 13 2.1 Eigenvalues................................ 13 2.2 Strongly Regular Graphs......................... 15 2.3 Intersection Numbers.......................... 16 2.4 Krein Parameters............................. 17 2.5 The Frame Quotient........................... 20 3 An Inner Product 23 3.1 An Inner Product............................. 23 3.2 Orthogonal Projection.......................... 24 3.3 Linear Programming........................... 25 3.4 Cliques and Cocliques.......................... 28 3.5 Feasible Automorphisms........................ 30 4 Products and Tensors 33 4.1 Kronecker Products............................ 33 4.2 Tensor Products.............................. 34 4.3 Tensor Powers............................... 37 4.4 Generalized Hamming Schemes.................... 38 4.5 A Tensor Identity............................. 39 v vi CONTENTS 4.6 Applications................................ 41 5 Subschemes
    [Show full text]
  • Group Divisible Association Scheme Let There Be V Treatments Which Can Be Represented As V = Pq
    Analysis of Variance and Design of Experiments-II MODULE - III LECTURE - 17 PARTIALLY BALANCED INCOMPLETE BLOCK DESIGN (PBIBD) Dr. Shalabh Department of Mathematics & Statistics Indian Institute of Technology Kanpur 2 Group divisible association scheme Let there be v treatments which can be represented as v = pq. Now divide the v treatments into p groups with each group having q treatments such that any two treatments in the same group are the first associates and the two treatments in different groups are the second associates. This is called the group divisible type scheme. The scheme simply amounts to arrange the v = pq treatments in a (p x q) rectangle and then the association scheme can be exhibited. The columns in the (p x q) rectangle will form the groups. Under this association scheme, nq1 = −1 n2 = qp( − 1), hence (q−+− 1)λλ12 qp ( 1) =− rk ( 1) and the parameters of second kind are uniquely determined by p and q. In this case qq−−20 0 1 PP12= , = , 0qp (− 1) q −−1 qp ( 2) For every group divisible design, r ≥ λ1, rk−≥ vλ2 0. 3 If r = λ1 , then the group group divisible design is said to be singular. Such singular group divisible design can always be derived from a corresponding BIBD. To obtain this, just replace each treatment by a group of q treatments. In general, if a BIBD has parameters bvrk*, *, *, *,λ *, then a divisible group divisible design is obtained which has following parameters b= b*, v = qv*, r = r*, k = qk*, λ12 = r, λλ =*, n 1 = p, n 2 = q.
    [Show full text]
  • ISMS 1993 2096T.Pdf
    Tht1 library ~ the Oepertmef't of St.at~ics North ~rolin8 St. University . , ASSOCIATION-BALANCED ARRAYS WITH APPLICATIONS TO EXPERIMENTAL DESIGN by Kamal Benchekroun A Dissertation submitted to the faculty of The Uni­ versity of North Carolina at Chapel Hill in partial fnlfiJ1rnent of the requirements for the degree of • Doctor of Philosophy in the Department of Statistics. Chapel Hill 1993 ~pproved by: J~L c~~v __L· Advisor KAMAL BENCHEKROUN. Association-Balanced Arrays with Applications to Experimental Design (Under the direction of INDRA M. CHAKRAVARTI.) • ABSTRACT This dissertation considers block designs for the comparison of v treatments where measurements from different blocks are uncorrelated and measurements in the same block have an arbitrary positive definite covariance matrix V, which is the same for all the blocks. Martin and Eccleston (1991) show that, for any V, a semi-balanced array of strength two, defined in Rao (1961, 1973), is universally optimal for the gen­ eralized least squares estimate of treatment effects over binary block designs, and weakly universally optimal for the ordinary least squares estimate over balanced incomplete block designs. The existence of these arrays requires a large number of columns (or blocks). The purpose of this dissertation is to introduce new series of • arrays relaxing this constraint and to discuss their performance as block designs. Based on the concept of association scheme, .an s-associate class association­ balanced array (or simply ABA) is defined, and some constructions are given. For any V, the variance matrix of the generalized (or the ordinary) least squares esti­ mate of treatment effects for an ABA is shown to be a constant multiple of that under the usual uncorrelated model, and a combinatorial characterization of the latter condition is given.
    [Show full text]
  • On Highly Regular Digraphs Oktay Olmez Iowa State University
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2012 On highly regular digraphs Oktay Olmez Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Olmez, Oktay, "On highly regular digraphs" (2012). Graduate Theses and Dissertations. 12626. https://lib.dr.iastate.edu/etd/12626 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On highly regular digraphs by Oktay Olmez A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Sung Yell Song, Major Professor Maria Axenovich Clifford Bergman L. Steven Hou Paul Sacks Iowa State University Ames, Iowa 2012 Copyright c Oktay Olmez, 2012. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my wife, Sevim, without whose support I would not have been able to complete this work. I would also like to thank my friends and family for their loving guidance during the writing of this work. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v ABSTRACT . vi CHAPTER 1. GENERAL OVERVIEW AND INTRODUCTION . 1 CHAPTER 2. PREMINILARIES . 4 2.1 Finite Incidence Structures . .4 2.2 Difference Sets . .7 1 2.3 1 2 -Designs . .9 2.4 Association Schemes .
    [Show full text]
  • Association Schemes. Designed Experiments, Algebra and Combinatorics,Byrose- Mary A
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 43, Number 2, Pages 249–253 S 0273-0979(05)01077-3 Article electronically published on October 7, 2005 Association schemes. Designed experiments, algebra and combinatorics,byRose- mary A. Bailey, Cambridge University Press, Cambridge, 2004, xvii+387 pp., US$70.00, ISBN 0-521-82446-X An association scheme (or a scheme as we shall say briefly) is a mathematical structure that has been created by statisticians [3], which, during the last three decades, has been intensively investigated by combinatorialists [2], and which, fi- nally, turned out to be an algebraic object generalizing in a particularly natural way the concept of a group [4], [8]. The definition of a scheme could easily catch the interest of an undergraduate student: given a set X, one calls a partition S of the cartesian product X × X a scheme on X if S satisfies the following three conditions: • The identity 1X := {(x, x) | x ∈ X} belongs to S. • For each element s in S, s∗ := {(y, z) | (z,y) ∈ s} belongs to S. • Let s be an element in S,andlet(y, z) be an element in s. Then, for any two elements p and q in S, the number of elements x with (y, x) ∈ p and (x, z) ∈ q does not depend on y or z,onlyons. The cardinal numbers arising from the last condition are usually denoted by apqs andcalledthestructure constants of S. The elements of S are defined to be subsets of X × X. This means that they are binary relations on X.
    [Show full text]
  • (Symmetric) Association Schemes Was First Given in the Design Of
    COMMUTATIVE ASSOCIATION SCHEMES WILLIAM J. MARTIN AND HAJIME TANAKA Abstract. Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of incep- tion, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the “commutative case,” has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, co- metric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group- theoretic symmetry, culminating in Schrijver’s SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger al- gebra in Delsarte Theory as a central topic for future work. 1. Introduction The concept of (symmetric) association schemes was first given in the design of experiments [27, 28]. It can also be viewed as a purely combinatorial generalization of the concept of finite transitive permutation groups.1 The Bose–Mesner algebra, which is a fundamental tool in the theory, was introduced in [26]. The monumental thesis of P. Delsarte [52] proclaimed the importance of commutative association schemes as a unifying framework for coding theory and design theory.
    [Show full text]
  • Design and Analysis of Experiments Volume 2
    Design and Analysis of Experiments Volum e 2 Advanced Experimental Design KLAUS HINKELMANN Virginia Polytechnic Institute and State University Department of Statistics Blacksburg, VA OSCAR KEMPTHORNE Iowa State University Department of Statistics Ames, IA A JOHN WILEY & SONS, INC., PUBLICATION Design and Analysis of Experiments Design and Analysis of Experiments Volum e 2 Advanced Experimental Design KLAUS HINKELMANN Virginia Polytechnic Institute and State University Department of Statistics Blacksburg, VA OSCAR KEMPTHORNE Iowa State University Department of Statistics Ames, IA A JOHN WILEY & SONS, INC., PUBLICATION Copyright 2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose.
    [Show full text]
  • CURRICULUM VITAE Xinggang Liu, M.D. Doctoral Candidate
    Determinants of Intrathoracic Adipose Tissue Volume and its Association with Cardiovascular Disease Risk Factors Item Type dissertation Authors Liu, Xinggang Publication Date 2012 Abstract Background: The volume of intrathoracic fat has been associated in some studies with cardiovascular diseases risk factors. To further assess the role of intrathoracic fat in coronary atherosclerosis risk, we measured the volume of intrathoracic fat i... Keywords intrathoracic adipose tissue; Cardiovascular Diseases; Genome- Wide Association Study; Obesity Download date 28/09/2021 02:45:03 Link to Item http://hdl.handle.net/10713/1666 CURRICULUM VITAE Xinggang Liu, M.D. Doctoral Candidate, Department of Epidemiology and Public Health University of Maryland School of Medicine CONTACT INFORMATION Epidemiology and Human Genetics Program School of Medicine University of Maryland, Baltimore 660 West Redwood St., Howard Hall 140C Baltimore, MD 21202 Phone: 14109005003 Email: [email protected] EDUCATION 2006–2012 Ph.D. in Epidemiology (expected) Epidemiology and human Genetics Program, School of Medicine, University of Maryland, Baltimore, MD 1999–2004 M.D. West China Medical Center, Sichuan University, Chengdu, Sichuan, China POSTGRADUATE TRAINING 2004–2006 Thoracic Surgical Resident Department of Thoracic Surgery, West China Hospital REASERCH TRAINING 2006–2012 Graduate Research Assistant Department of Epidemiology and Preventive Medicine, University of Maryland, Baltimore, MD TEACHING SERVICE 2010 Teaching Assistant Clinical Trials and Experimental Design,
    [Show full text]
  • Uniform Mixing of Quantum Walks and Association Schemes
    CORE Metadata, citation and similar papers at core.ac.uk Provided by University of Waterloo's Institutional Repository UNIFORM MIXING OF QUANTUM WALKS AND ASSOCIATION SCHEMES by Natalie E. Mullin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2013 c Natalie E. Mullin 2013 AUTHOR'S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In recent years quantum algorithms have become a popular area of mathe- matical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous- time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quan- tum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times.
    [Show full text]
  • (Statistics) Revised Syllabus Wef 2013-2014
    UNIVERSITY OF MADRAS DEPARTMENT OF STATISTICS M.Sc (Statistics) Revised Syllabus w.e.f. 2013-2014 1 STATISTICS (Course Proposals for the academic year 2013 – 2014) A – CORE COURSES Subject Code Title of the Course C/E/S L T P C I SEMESTER MSI C121 Statistical Mathematics C 3 1 0 4 MSI C122 Measure and Probability Theory C 3 1 0 4 MSI C123 Distribution Theory C 3 1 0 4 MSI C124 R Programming C 3 1 0 4 Elective 1 E 2 1 0 3 Elective 2 E 2 1 0 3 Soft Skill S 2 II SEMESTER MSI C125 Linear Regression Analysis C 3 1 0 4 MSI C126 Sampling Theory C 3 1 0 4 MSI C127 Statistical Estimation Theory C 3 1 0 4 MSI C128 Statistical Laboratory-I C 0 0 2 2 Elective 3 E 2 1 0 3 Elective 4 E 2 1 0 3 Soft Skill S 2 III SEMESTER MSI C129 Multivariate Analysis C 3 1 0 4 MSI C130 Testing Statistical Hypotheses C 3 1 0 4 MSI C131 Design & Analysis of Experiments C 3 1 0 4 Elective 5 E 2 1 0 3 Elective 6 E 2 1 0 3 Soft Skill S 2 Internship I 2 IV SEMESTER MSI C132 Statistical Quality Management C 3 1 0 4 MSI C133 Advanced Operations Research C 3 1 0 4 MSI C134 Statistical Laboratory-II C 0 0 2 2 MSI C135 Statistical Software Practical C 0 0 2 2 MSI C136 Project Work / Dissertation C 0 6 0 6 Elective 7 E 2 1 0 3 Soft Skill S 2 2 B – ELECTIVE COURSES : Subject Code Title of the Course L T P C MSI E121 Actuarial Statistics 3 0 0 3 MSI E122 Statistical Methods for Epidemiology 3 0 0 3 MSI E123 Stochastic Processes 3 0 0 3 MSI E 124 Non -Parametric Inference 3 0 0 3 MSI E125 Data Mining 3 0 0 3 MSI E126 Bayesian Inference 3 0 0 3 MSI E127 Reliability Theory 3 0 0 3 MSI E128 Survival Analysis 3 0 0 3 MSI E129 Categorical Data Analysis 3 0 0 3 MSI E130 * Bio -Statistics 3 0 0 3 * TO OTHER DEPARTMENTS ONLY 3 MSI C121 Statistical Mathematics C 3 1 0 4 Pre-requisite: Undergraduate level Mathematics.
    [Show full text]
  • Ph.D. (Statistics)
    Curriculum for Doctor of Philosophy Programme in Statistics Program Code: 177 Department of Statistics Indira Gandhi National Tribal University Amarkantak, India 1 Contents 1 PhD(Statistics) Program 3 1.1 Qualifying Examination Requirements . 3 1.2 Syllabus for Qualifying Examination . 4 2 Courses Offered in PhD(Statistics) Program 5 2.1 Mandatory Core Courses . 5 2.2 Discipline Specific Core Courses . 5 2.3 Semester-wise Structure of the Programme . 6 3 Detailed Syllabus 7 3.1 SCC-01 Research Methodologies . 7 3.2 SCC-02 Lab . 8 3.3 RSTAT-01 Biostatistics . 8 3.4 RSTAT-02 Analysis of Biostatistical Data . 9 3.5 RSTAT-03 Advanced Design of Sample Surveys . 10 3.6 RSTAT-04 Advanced Design and Analysis of Experiments . 11 3.7 RSTAT-05 Testing of Hypothesis . 12 3.8 RSTAT-06 Generalised Linear Models and Inference . 13 3.9 RSTAT-07: Bayesian Inference . 14 3.10 RSTAT-08: Theory of Estimation . 15 3.11 RSTAT-09: Operations Research : Stochastic and Determin- istic Models and Optimization . 16 2 1 PhD(Statistics) Program 1.1 Qualifying Examination Requirements Subject to the University regulations, the following shall be applicable to all students joining the PhD program in the Department of Statistics: 1. Each student in the Ph.D. programme in the Department of Statistics is required to apply for PhD Qualifying Examination. 2. The qualifying examination, if required (in case of non - NET can- didates) shall consist of two sections namely Part - A (Research Aptitude Test) and Part - B (Statistics). Part- A and Part - B shall consist of 30 and 70 objective type questions respectively.
    [Show full text]