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Additive Combinatorics
Additive Combinatorics Summer School, Catalina Island ∗ August 10th - August 15th 2008 Organizers: Ciprian Demeter, IAS and Indiana University, Bloomington Christoph Thiele, University of California, Los Angeles ∗supported by NSF grant DMS 0701302 1 Contents 1 On the Erd¨os-Volkmann and Katz-Tao Ring Conjectures 5 JonasAzzam,UCLA .......................... 5 1.1 Introduction............................ 5 1.2 The Ring, Distance, and Furstenburg Conjectures . 5 1.3 MainResults ........................... 7 2 Quantitative idempotent theorem 10 YenDo,UCLA ............................. 10 2.1 Introduction............................ 10 2.2 Themainargument........................ 11 2.3 Proofoftheinductionstep. 12 2.4 Construction of the required Bourgain system . 15 3 A Sum-Product Estimate in Finite Fields, and Applications 21 JacobFox,Princeton . .. .. 21 3.1 Introduction............................ 21 3.2 Preliminaries ........................... 23 3.3 ProofoutlineofTheorem1. 23 4 Growth and generation in SL2(Z/pZ) 26 S.ZubinGautam,UCLA. 26 4.1 Introduction............................ 26 4.2 Outlineoftheproof. .. .. 27 4.3 Part(b)frompart(a) ...................... 28 4.4 Proofofpart(a) ......................... 29 4.4.1 A reduction via additive combinatorics . 29 4.4.2 Tracesandgrowth . 30 4.4.3 A reduction to additive combinatorics . 31 4.5 Expandergraphs ......................... 32 4.6 Recentfurtherprogress. 33 5 The true complexity of a system of linear equations 35 DerrickHart,UCLA .......................... 35 5.1 Introduction............................ 35 5.2 Quadratic fourier analysis and initial reductions . .... 37 5.3 Dealing with f1 .......................... 39 2 5.4 Finishingtheproofofthemaintheorem . 42 6 On an Argument of Shkredov on Two-Dimensional Corners 43 VjekoslavKovaˇc,UCLA . 43 6.1 Somehistoryandthemainresult . 43 6.2 Outlineoftheproof. .. .. 44 6.3 Somenotation........................... 45 6.4 Mainingredientsoftheproof . 46 6.4.1 Generalized von Neumann lemma . -
On the Number of Cubic Orders of Bounded Discriminant Having
On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman August 8, 2013 Abstract For a binary quadratic form Q, we consider the action of SOQ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style count- ing problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automor- phism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant D, the shape is equidistributed in the class group ClD of binary quadratic forms of discriminant D. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field. -
“For Christ and Covenant:” a Movement for Equal Consideration in Early Nineteenth Century Nova Scotia
“For Christ and Covenant:” A Movement for Equal Consideration in Early Nineteenth Century Nova Scotia. By Holly Ritchie A Thesis Submitted to Saint Mary’s University, Halifax, Nova Scotia in Partial Fulfilment of the Requirements for the Degree of Master of Arts in History © Copyright Holly Ritchie, 2017 November, 2017, Halifax, Nova Scotia Approved: Dr. S. Karly Kehoe Supervisor Approved: Dr. John Reid Reader Approved: Dr. Jerry Bannister Examiner Date: 30 November 2017 1 Abstract “For Christ and Covenant:” A Movement for Equal Consideration in Early Nineteenth Century Nova Scotia. Holly Ritchie Reverend Dr. Thomas McCulloch is a well-documented figure in Nova Scotia’s educational historiography. Despite this, his political activism and Presbyterian background has been largely overlooked. This thesis offers a re-interpretation of the well-known figure and the Pictou Academy’s fight for permeant pecuniary aid. Through examining Scotland’s early politico-religious history from the Reformation through the Covenanting crusades and into the first disruption of the Church of Scotland, this thesis demonstrates that the language of political disaffection was frequently expressed through the language of religion. As a result, this framework of response was exported with the Scottish diaspora to Nova Scotia, and used by McCulloch to stimulate a movement for equal consideration within the colony. Date: 30 November 2017 2 Acknowledgements Firstly, to the wonderful Dr. S. Karly Kehoe, thank you for providing me with an opportunity beyond my expectations. A few lines of acknowledgement does not do justice to the impact you’ve had on my academic work, and my self-confidence. -
Collision Times of Random Walks and Applications to the Brownian
Collision times of random walks and applications to the Brownian web David Coupier1, Kumarjit Saha2, Anish Sarkar3, and Viet Chi Tran4 1Univ. de Valenciennes, CNRS 2Ashoka University 3Indian Statistical Institute, Delhi 4Univ. Lille, CNRS February 12, 2019 Abstract Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their conver- gence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length t(> 0), all starting within a segment of length ε, is of small order of ε. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first col- lision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]). -
The Church Militant: the American Loyalist Clergy and the Making of the British Counterrevolution, 1701-92
The Church Militant: The American Loyalist Clergy and the Making of the British Counterrevolution, 1701-92 Peter W. Walker Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2016 © 2016 Peter Walker All rights reserved ABSTRACT The Church Militant: The American Loyalist Clergy and the Making of the British Counterrevolution, 1701-92 Peter W. Walker This dissertation is a study of the loyalist Church of England clergy in the American Revolution. By reconstructing the experience and identity of this largely-misunderstood group, it sheds light on the relationship between church and empire, the role of religious pluralism and toleration in the American Revolution, the dynamics of loyalist politics, and the religious impact of the American Revolution on Britain. It is based primarily on the loyalist clergy’s own correspondence and writings, the records of the American Loyalist Claims Commission, and the archives of the SPG (the Church of England’s missionary arm). The study focuses on the New England and Mid-Atlantic colonies, where Anglicans formed a religious minority and where their clergy were overwhelmingly loyalist. It begins with the founding of the SPG in 1701 and its first forays into America. It then examines the state of religious pluralism and toleration in New England, the polarising contest over the proposed creation of an American bishop after the Seven Years’ War, and the role of the loyalist clergy in the Revolutionary War itself, focusing particularly on conflicts occasioned by the Anglican liturgy and Book of Common Prayer. -
MAT 362 at Stony Brook, Spring 2011
MAT 362 Differential Geometry, Spring 2011 Instructors' contact information Course information Take-home exam Take-home final exam, due Thursday, May 19, at 2:15 PM. Please read the directions carefully. Handouts Overview of final projects pdf Notes on differentials of C1 maps pdf tex Notes on dual spaces and the spectral theorem pdf tex Notes on solutions to initial value problems pdf tex Topics and homework assignments Assigned homework problems may change up until a week before their due date. Assignments are taken from texts by Banchoff and Lovett (B&L) and Shifrin (S), unless otherwise noted. Topics and assignments through spring break (April 24) Solutions to first exam Solutions to second exam Solutions to third exam April 26-28: Parallel transport, geodesics. Read B&L 8.1-8.2; S2.4. Homework due Tuesday May 3: B&L 8.1.4, 8.2.10 S2.4: 1, 2, 4, 6, 11, 15, 20 Bonus: Figure out what map projection is used in the graphic here. (A Facebook account is not needed.) May 3-5: Local and global Gauss-Bonnet theorem. Read B&L 8.4; S3.1. Homework due Tuesday May 10: B&L: 8.1.8, 8.4.5, 8.4.6 S3.1: 2, 4, 5, 8, 9 Project assignment: Submit final version of paper electronically to me BY FRIDAY MAY 13. May 10: Hyperbolic geometry. Read B&L 8.5; S3.2. No homework this week. Third exam: May 12 (in class) Take-home exam: due May 19 (at presentation of final projects) Instructors for MAT 362 Differential Geometry, Spring 2011 Joshua Bowman (main instructor) Office: Math Tower 3-114 Office hours: Monday 4:00-5:00, Friday 9:30-10:30 Email: joshua dot bowman at gmail dot com Lloyd Smith (grader Feb. -
On the Number of Cubic Orders of Bounded Discriminant Having Automorphism Group C3, and Related Problems Manjul Bhargava and Ariel Shnidman
Algebra & Number Theory Volume 8 2014 No. 1 On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman msp ALGEBRA AND NUMBER THEORY 8:1 (2014) msp dx.doi.org/10.2140/ant.2014.8.53 On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems Manjul Bhargava and Ariel Shnidman For a binary quadratic form Q, we consider the action of SOQ on a 2-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegen- erate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymp- totic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. -
Optimization of the Manufacturing Process for Sheet Metal Panels Considering Shape, Tessellation and Structural Stability Optimi
Institute of Construction Informatics, Faculty of Civil Engineering Optimization of the manufacturing process for sheet metal panels considering shape, tessellation and structural stability Optimierung des Herstellungsprozesses von dünnwandigen, profilierten Aussenwandpaneelen by Fasih Mohiuddin Syed from Mysore, India A Master thesis submitted to the Institute of Construction Informatics, Faculty of Civil Engineering, University of Technology Dresden in partial fulfilment of the requirements for the degree of Master of Science Responsible Professor : Prof. Dr.-Ing. habil. Karsten Menzel Second Examiner : Prof. Dr.-Ing. Raimar Scherer Advisor : Dipl.-Ing. Johannes Frank Schüler Dresden, 14th April, 2020 Task Sheet II Task Sheet Declaration III Declaration I confirm that this assignment is my own work and that I have not sought or used the inadmissible help of third parties to produce this work. I have fully referenced and used inverted commas for all text directly quoted from a source. Any indirect quotations have been duly marked as such. This work has not yet been submitted to another examination institution – neither in Germany nor outside Germany – neither in the same nor in a similar way and has not yet been published. Dresden, Place, Date (Signature) Acknowledgement IV Acknowledgement First, I would like to express my sincere gratitude to Prof. Dr.-Ing. habil. Karsten Menzel, Chair of the "Institute of Construction Informatics" for giving me this opportunity to work on my master thesis and believing in my work. I am very grateful to Dipl.-Ing. Johannes Frank Schüler for his encouragement, guidance and support during the course of work. I would like to thank all the staff from "Institute of Construction Informatics " for their valuable support. -
The Globalization of K-Pop: the Interplay of External and Internal Forces
THE GLOBALIZATION OF K-POP: THE INTERPLAY OF EXTERNAL AND INTERNAL FORCES Master Thesis presented by Hiu Yan Kong Furtwangen University MBA WS14/16 Matriculation Number 249536 May, 2016 Sworn Statement I hereby solemnly declare on my oath that the work presented has been carried out by me alone without any form of illicit assistance. All sources used have been fully quoted. (Signature, Date) Abstract This thesis aims to provide a comprehensive and systematic analysis about the growing popularity of Korean pop music (K-pop) worldwide in recent years. On one hand, the international expansion of K-pop can be understood as a result of the strategic planning and business execution that are created and carried out by the entertainment agencies. On the other hand, external circumstances such as the rise of social media also create a wide array of opportunities for K-pop to broaden its global appeal. The research explores the ways how the interplay between external circumstances and organizational strategies has jointly contributed to the global circulation of K-pop. The research starts with providing a general descriptive overview of K-pop. Following that, quantitative methods are applied to measure and assess the international recognition and global spread of K-pop. Next, a systematic approach is used to identify and analyze factors and forces that have important influences and implications on K-pop’s globalization. The analysis is carried out based on three levels of business environment which are macro, operating, and internal level. PEST analysis is applied to identify critical macro-environmental factors including political, economic, socio-cultural, and technological. -
A Computational Analysis of Information Structure Using Parallel Expository Texts in English and Japanese
University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science January 1999 A Computational Analysis of Information Structure Using Parallel Expository Texts in English and Japanese Nobo N. Komagata University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/ircs_reports Komagata, Nobo N., "A Computational Analysis of Information Structure Using Parallel Expository Texts in English and Japanese" (1999). IRCS Technical Reports Series. 46. https://repository.upenn.edu/ircs_reports/46 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-99-07. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/46 For more information, please contact [email protected]. A Computational Analysis of Information Structure Using Parallel Expository Texts in English and Japanese Abstract This thesis concerns the notion of 'information structure': informally, organization of information in an utterance with respect to the context. Information structure has been recognized as a critical element in a number of computer applications: e.g., selection of contextually appropriate forms in machine translation and speech generation, and analysis of text readability in computer-assisted writing systems. One of the problems involved in these applications is how to identify information structure in extended texts. This problem is often ignored, assumed to be trivial, or reduced to a sub-problem that does not correspond to the complexity of realistic texts. A handful of computational proposals face the problem directly, but they are generally limited in coverage and all suffer from lack of evaluation. To fully demonstrate the usefulness of information structure, it is essential to apply a theory of information structure to the identification problem and to provide an evaluation method. -
ASIAN Athletics 2 0 1 7 R a N K I N G S
ASIAN athletics 2 0 1 7 R a n k i n g s compiled by: Heinrich Hubbeling - ASIAN AA Statistician – C o n t e n t s Page 1 Table of Contents/Abbreviations for countries 2 - 3 Introduction/Details 4 - 9 Asian Continental Records 10 - 60 2017 Rankings – Men events 60 Name changes (to Women´s Rankings) 61 - 108 2017 Rankings – Women events 109 – 111 Asian athletes in 2017 World lists 112 Additions/Corrections to 2016 Rankings 113 - 114 Contacts for other publications etc. ============================================================== Abbreviations for countries (as used in this booklet) AFG - Afghanistan KGZ - Kyrghizstan PLE - Palestine BAN - Bangladesh KOR - Korea (South) PRK - D P R Korea BHU - Bhutan KSA - Saudi Arabia QAT - Qatar BRN - Bahrain KUW - Kuwait SGP - Singapore BRU - Brunei LAO - Laos SRI - Sri Lanka CAM - Cambodia LBN - Lebanon SYR - Syria CHN - China MAC - Macau THA - Thailand HKG - Hongkong MAS - Malaysia TJK - Tajikistan INA - Indonesia MDV - Maldives TKM - Turkmenistan IND - India MGL - Mongolia TLS - East Timor IRI - Iran MYA - Myanmar TPE - Chinese Taipei IRQ - Iraq NEP - Nepal UAE - United Arab E. JOR - Jordan OMA - Oman UZB - Uzbekistan JPN - Japan PAK - Pakistan VIE - Vietnam KAZ - Kazakhstan PHI - Philippines YEM - Yemen ============================================================== Cover Photo: MUTAZ ESSA BARSHIM -World Athlet of the Year 2017 -World Champion 2017 -World 2017 leader with 2.40 m (achieved twice) -undefeated during the 2017 season 1 I n t r o d u c t i o n With this booklet I present my 29th consecutive edition of Asian athletics statistics. As in the previous years I am very grateful to the ASIAN ATHLETICS ASSOCIATION and its secretary and treasurer, Mr Maurice Nicholas as well as to Mrs Regina Long; without their support I would not have been able to realise this project. -
A Correspondence Principle for the Gowers Norms 1 Introduction
Journal of Logic & Analysis 4:4 (2012) 1–11 1 ISSN 1759-9008 A correspondence principle for the Gowers norms HENRY TOWSNER Abstract: The Furstenberg Correspondence shows that certain “local behavior” of dynamical system is equivalent to the behavior of sufficiently large finite systems. The Gowers uniformity norms, however, are not local in the relevant sense. We give a modified correspondence in which the Gowers norm is preserved. This extends to the integers a similar result by Tao and Zielger on finite fields. 2010 Mathematics Subject Classification 03H05, 37A05, 11B30 (primary) Keywords: Furstenberg correspondence, Gowers uniformity norms, nonstandard analysis 1 Introduction Informally speaking, the Furstenberg Correspondence [4, 5] shows that the “local behavior” of a dynamical system is controlled by the behavior of sufficiently large finite systems. By the local behavior of a dynamical system (X; B; µ, G), we mean the properties which can be stated using finitely many actions of G and the integral given by µ1. By a finite system, we just mean (S; P(S); c; G) where G is a infinite group, S jAj is a finite quotient of G, and c is the counting measure c(A):= jSj . The most well known example of such a property is the ergodic form of Szemeredi’s´ Theorem: For every k, every > 0, and every L1 function f , if R fdµ ≥ then R Qk−1 −jn there is some n such that j=0 T fdµ > 0. The Furstenberg Correspondence shows that this is equivalent to the following statement of Szemeredi’s´ Theorem: For every k and every > 0, there is an N and a δ > 0 such that if m ≥ N and f : [0; m − 1] ! [−1; 1] is such that R fdc ≥ then there is some n R Qk−1 such that j=0 f (x + jn)dc(x) ≥ δ.