Abstract Algebra: Monoids, Groups, Rings

Total Page:16

File Type:pdf, Size:1020Kb

Abstract Algebra: Monoids, Groups, Rings Notes on Abstract Algebra John Perry University of Southern Mississippi [email protected] http://www.math.usm.edu/perry/ Copyright 2009 John Perry www.math.usm.edu/perry/ Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States You are free: to Share—to copy, distribute and transmit the work • to Remix—to adapt the work Under• the following conditions: Attribution—You must attribute the work in the manner specified by the author or licen- • sor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial—You may not use this work for commercial purposes. • Share Alike—If you alter, transform, or build upon this work, you may distribute the • resulting work only under the same or similar license to this one. With the understanding that: Waiver—Any of the above conditions can be waived if you get permission from the copy- • right holder. Other Rights—In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights; ◦ Apart from the remix rights granted under this license, the author’s moral rights; ◦ Rights other persons may have either in the work itself or in how the work is used, ◦ such as publicity or privacy rights. Notice—For any reuse or distribution, you must make clear to others the license terms of • this work. The best way to do this is with a link to this web page: http://creativecommons.org/licenses/by-nc-sa/3.0/us/legalcode i Table of Contents Reference sheet for notation ...........................................................v A few acknowledgements .............................................................vii Preface ..............................................................................viii Overview ..........................................................................viii Three interesting problems ............................................................1 Part . Monoids 1. From integers to monoids ...........................................................4 1. Some facts about the integers .........................................................5 2. Integers, monomials, and monoids ...................................................11 3. Direct Products and Isomorphism ....................................................15 Part I. Groups 2. Groups ............................................................................22 1. Groups ...........................................................................22 2. The symmetries of a triangle.........................................................29 3. Cyclic groups and order ............................................................36 4. Elliptic Curves ....................................................................43 3. Subgroups .........................................................................47 1. Subgroups.........................................................................47 2. Cosets ............................................................................51 3. Lagrange’s Theorem ................................................................56 4. Quotient Groups ..................................................................59 5. “Clockwork” groups................................................................64 6. “Solvable” groups ..................................................................67 4. Isomorphisms ......................................................................72 1. From functions to isomorphisms .....................................................72 2. Consequences of isomorphism .......................................................78 3. The Isomorphism Theorem ..........................................................83 4. Automorphisms and groups of automorphisms.........................................87 5. Groups of permutations............................................................92 1. Permutations......................................................................92 2. Groups of permutations ...........................................................101 3. Dihedral groups ..................................................................103 4. Cayley’s Theorem .................................................................108 5. Alternating groups ................................................................111 ii 6. The 15-puzzle ....................................................................116 6. Number theory ...................................................................119 1. GCD and the Euclidean Algorithm .................................................119 2. The Chinese Remainder Theorem ...................................................123 3. Multiplicative clockwork groups ....................................................130 4. Euler’s Theorem ..................................................................136 5. RSA Encryption ..................................................................140 Part II. Rings 7. Rings .............................................................................147 1. A structure for addition and multiplication ..........................................147 2. Integral Domains and Fields .......................................................150 3. Polynomial rings .................................................................155 4. Euclidean domains................................................................163 8. Ideals .............................................................................169 1. Ideals............................................................................169 2. Principal Ideals ...................................................................175 3. Prime and maximal ideals .........................................................178 4. Quotient Rings ...................................................................181 5. Finite Fields I ....................................................................186 6. Ring isomorphisms ...............................................................192 7. A generalized Chinese Remainder Theorem ..........................................197 8. Nullstellensatz....................................................................200 9. Rings and polynomial factorization................................................203 1. The link between factoring and ideals ...............................................203 2. Unique Factorization domains .....................................................206 3. Finite fields II ....................................................................209 4. Polynomial factorization in finite fields ..............................................214 5. Factoring integer polynomials ......................................................219 10. Gröbner bases ...................................................................224 1. Gaussian elimination ............................................................225 2. Monomial orderings .............................................................231 3. Matrix representations of monomial orderings ......................................238 4. The structure of a Gröbner basis ...................................................241 5. Buchberger’s algorithm ...........................................................251 6. Elementary applications ..........................................................259 11. Advanced methods of computing Gröbner bases ..................................264 1. The Gebauer-Möller algorithm ....................................................264 2. The F4 algorithm ................................................................273 iii 3. Signature-based algorithms to compute a Gröbner basis...............................278 Part III. Appendices Where can I go from here? ...........................................................287 Advanced group theory .............................................................287 Advanced ring theory ..............................................................287 Applications.......................................................................287 Hints to Exercises ...................................................................288 Hints to Chapter1 .................................................................288 Hints to Chapter2 .................................................................288 Hints to Chapter3 .................................................................290 Hints to Chapter4 .................................................................291 Hints to Chapter5 .................................................................291 Hints to Chapter6 .................................................................292 Hints to Chapter7 .................................................................293 Hints to Chapter8 .................................................................294 Hints to Chapter9 .................................................................295 Hints to Chapter 10 ................................................................295 Index................................................................................296 References...........................................................................300 iv Reference sheet for notation [r ] the element r + nZ of Zn g the group (or ideal) generated by g h i A3 the alternating group on three elements A/ G for G a group, A is a normal subgroup of G A/ R for R a ring, A is an ideal of R C the complex numbers a + b i : a, b C and i = p 1 [G,G] commutator subgroupf of a group
Recommended publications
  • 5Efa37f24b66fa5fa397c116 Bec
    © Prolego, Inc., 2018. All rights reserved. ISBN 978-0-692-19233-7 This book or parts thereof may not be reproduced in any form, stored in any retrieval system, or transmitted in any form by any means—electronic, mechanical, photocopy, recording, or otherwise—without prior written permission of Prolego, except as provided by United States of America copyright law. For permission requests contact Prolego. Prolego, Inc. 400 West Peachtree Street NW Suite #4 - 868 Atlanta, GA 30308 prolego.io [email protected] Contents - 07 Introduction - 23 PART 1: AI Fundamentals - 45 PART 2: Discovering AI Opportunities - 75 PART 3: Building a Winning AI Strategy - 97 PART 4: Launching Your First AI Product - 117 Epilogue - 121 Appendices 6 – INTRODUCTION 6 – INTRODUCTION Even if there were a trustworthy way to send money over the Internet– which there isn’t–the network is missing a most essential ingredient of capitalism: salespeople. — Clifford Stoll “The Internet? Bah!” Newsweek 1995 BECOME AN AI COMPANY IN 90 DAYS – 7 Introduction THE ERA OF INTELLIGENT COMPUTING Congratulations. You’re living through a once-in-a-generation technology shift—the era of artificial intelligence, or AI. Like previous fundamental shifts such as electricity, the computer, or the Internet, AI will change everything. Skeptical about such big claims? I don’t blame you. Over the past few decades we’ve been bombarded by an endless parade of new technologies promising to “disrupt” everything and solve all our problems. BECOME AN AI COMPANY IN 90 DAYS 8 – INTRODUCTION Most of these technologies never yield more than incremental benefits: > Object-oriented programming made software only slightly more reusable.
    [Show full text]
  • EG Analysistofairhousing Cover Draftmay2009.Ai
    Analysis of Impediments to Fair Housing & Fair Housing Plan - May , C ITY OF ELK G ROVE A NALYSIS OF I MPEDIMENTS TO F AIR H OUSING C HOICE FAIR HOUSING PLAN 2008-09 Prepared by: CITY OF ELK GROVE 8401 LAGUNA PALMS WAY ELK GROVE, CA 95758 MAY 15, 2009 TABLE OF CONTENTS TABLE OF CONTENTS Executive Summary .................................................................................................................................. ES-1 1 Introduction ........................................................................................................................................ 1-1 2 Review of Previous Analysis of Impediments ................................................................................. 2-1 3 Community Setting ............................................................................................................................ 3-1 4 Housing Market Conditions .............................................................................................................. 4-1 5 Mortgage Lending (HMDA Data) ................................................................................................... 5-1 6 Government Barriers to Fair Housing Choice ................................................................................ 6-1 7 Fair Housing Enforcement and Education ..................................................................................... 7-1 8 Summary Analysis .............................................................................................................................
    [Show full text]
  • Analysis of Impediments to Fair Housing Choice in New Hampshire
    Analysis of Impediments to Fair Housing Choice in New Hampshire 2020 UPDATE May 2021 Prepared for New Hampshire Housing Finance Authority and New Hampshire Community Development Finance Authority by New Hampshire Legal Assistance Analysis of Impediments to Fair Housing Choice in New Hampshire 2020 Update May 2021 Prepared for New Hampshire Housing Finance Authority and New Hampshire Community Development Finance Authority by New Hampshire Legal Assistance The publication of the 2020 Update to the Analysis of Impediments to Fair Housing Choice in New Hampshire (2020 AI) marks the fifth Analysis conducted since 1996 when the Office of State Planning issued the original Analysis of Impediments (AI). By 2004, New Hampshire Housing Finance Authority (NHHFA) had taken responsibility for evaluating barriers to housing opportunity and published an update to the AI. In 2010, New Hampshire Community Development Finance Authority (CDFA) also began supporting publication of the AI. The most recent update was published in 2015. New Hampshire Legal Assistance (NHLA), under contract with NHHFA and CDFA, has produced all four of the updates. Scope of Investigation In preparing the 2020 AI Report, the following activities were undertaken: 1. Thirty-year review of affirmatively furthering fair housing in the state 2. Demographic analyses; 3. Evaluation of areas of high poverty and racial/ethnic concentrations; 4. A focus group and a set of one-on-one interviews with community members; 5. A survey of the state’s Public Housing Authorities (PHAs); 6. Analysis of the housing barriers facing residents with serious mental illness; 7. Research into legal developments at the federal and state level; 8.
    [Show full text]
  • 2Wsm1sbirsdt WBEUVEII the Lieu
    ; To! ~~. Rider life-saving team drowns in Maurer pool (8ee p. 27) .2wsM1SbiRsdT WBEUVEII THE lIEU. If. IS 'APIIIL I, 1912· 'Rider deals with moose on loose clean up the path of damaae By JIMMY OLSEN toro d~ by their c:ompet!.­ sure save OD au," er tumiD. the e&mpas iDto . : To add to the seasonal tioo, but we know better. and litter the l.rae berbi­ zoo. problem of Canada geese It', those damn meese." vores bave left 00 campus, He said he w.. astiDg for 00 campus, a herd said co-fouoder Sbarkey volUDteen to dress up .. roostia, The large animals ' have " Actually, that'd be re­ of moose hal settled 00 Wook. moose and " live with the also ambled. into classrooms dundant," said ,;ock Rhino, camp~ , said Godzi1Ia Belcb, jlerd, leani their ~ . " assistant director of Public and begwi to bellow. Roger Brooke Taney, assoc­ usociate director of the Phy­ If' that faU., said Taney, InformatioD. " It already is a iate dean of studeots, said .leaI Plant. the CoHe,. may also COIlIld· ZOO , " li lt's - very atlDoyiD.," said the College w.. colUidering sophomore Grinch McChurll­ biring reIlOWD~ bUDter Joe Belch said t be aDlmal. bobiDstilly. 'They ...ake you have diJrupted tnffic, intra­ . "BuO" W1Dtle to emrmiDate up," the beasts. m~ J&;mel a.a.d campaips Students arrested for Student GoYemmeJlt Au­ McChurliboo.i:llly. a ociation ~_. " On . the other band, we By WALD'- DAIIIIII!rr echoed ac:rosa campus, said· 6.aance major, said be aDd a mi,bt try to d.omesticate ApproJdrDately 100 ·studeD.
    [Show full text]
  • AI Watch Historical Evolution of Artificial Intelligence
    JRC TECHNICAL REPORTS AI Watch Historical Evolution of Artificial Intelligence Analysis of the three main paradigm shifts in AI EUR 30221 EN This publication is a Technical report by the Joint Research Centre (JRC), the European Commission’s science and knowledge service. It aims to provide evidence-based scientific support to the European policymaking process. The scientific output expressed does not imply a policy position of the European Commission. Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use that might be made of this publication. Contact information Blagoj DELIPETREV European Commission, Joint Research Centre, TP262, Via Fermi, 21027 Ispra (VA), ITALY [email protected] Tel.: + 39 0332 786 352 AI-Watch functional mailbox [email protected] EU Science Hub https://ec.europa.eu/jrc JRC120469 EUR 30221 EN PDF ISBN 978-92-76-18940-4 ISSN 1831-9424 doi:10.2760/801580 Luxembourg: Publications Office of the European Union, 2020 © European Union, 2020 The reuse policy of the European Commission is implemented by the Commission Decision 2011/833/EU of 12 December 2011 on the reuse of Commission documents (OJ L 330, 14.12.2011, p. 39). Except otherwise noted, the reuse of this document is authorised under the Creative Commons Attribution 4.0 International (CC BY 4.0) licence (https://creativecommons.org/licenses/by/4.0/). This means that reuse is allowed provided appropriate credit is given and any changes are indicated. For any use or reproduction of photos or other material that is not owned by the EU, permission must be sought directly from the copyright holders.
    [Show full text]
  • Aistrom – a Roadmap for Developing a Successful AI Strategy Arxiv
    aiSTROM { A roadmap for developing a successful AI strategy Dorien Herremans1 1Singapore University of Technology and Design June 2021 A total of 34% of AI research and development projects fails or are abandoned, accord- ing to a recent survey by Rackspace Technology of 1,870 companies (Rackspace Technol- ogy, 2021). We propose a new strategic framework, aiSTROM, that empowers managers to create a successful AI strategy based on a thorough literature review. This provides a unique and integrated approach that guides managers and lead developers through the var- ious challenges in the implementation process. In the aiSTROM framework, we start by identifying the top n potential projects (typically 3-5). For each of those, seven areas of focus are thoroughly analysed. These areas include creating a data strategy that takes into account unique cross-departmental machine learning data requirements, security, and legal requirements. aiSTROM then guides managers to think about how to put together an inter- disciplinary artificial intelligence (AI) implementation team given the scarcity of AI talent. Once an AI team strategy has been established, it needs to be positioned within the organi- zation, either cross-departmental or as a separate division. Other considerations include AI as a service (AIaas), or outsourcing development. Looking at new technologies, we have to consider challenges such as bias, legality of black-box-models, and keeping humans in the loop. Next, like any project, we need value-based key performance indicators (KPIs) to track and validate the progress. Depending on the company’s risk-strategy, a SWOT analysis (strengths, weaknesses, opportunities, and threats) can help further classify the shortlisted projects.
    [Show full text]
  • Alfakabi's Theory of State
    ALFAKABI'S THEORY OF STATE ISSIRTATION SUBMITTED FOR THE DEQREB OF MASTER OF PHILOSOPHY IN Political Science BY Saiyid Ali Sajjad Under the Supervision of Mr. Abdul Shafiq Beg READER DEPARTMENT OF POLITICAL SCIENCE ALIGARH MUSLIM UNIVERSITY ALIGARH 1980 1^ umiEn'o\'^''' DS260 •JX& CHECSED- PRBFACE So tax, there h^ive iaeon no d«tadLXed theniatic studLLea o£ trie ix>Utlcal philosophy o£ ALTAaAQI^ *rh« sacond Toach«r*' (al CHialllm al than!) oi. the Arab worlds Aristotle beinj tlie Firat. Whatever little viock is dofie« is in the field o£ his metaphysical* logiical and philosof^iical* or at most psychological conoapts only, in this study X have attsmptod to fonnulate a systematic theory o£ state wi the basis o£ Alfarabi*a own writings in Arabic, in doing so X h^ivm had to travel into the reijicms of metaip^ysicB, philoso{^y« religion* ethics as well as that of psychology as they are all mixed up and intricately woven into his political thoi^ht, AS a matter of fact these are the various baswsi of his tli^ory of state* t^iich cme oarmot ignore in a detailed and ccs^n^iMMiisive study of his ccmcept of the State and political authority. The present researdi project has h^en udertaken to supplicate for the M.phil, degree from Aligarh Muslim University, wliich 1 interU to further elaborate* and compare Alfarabl^s political thou;jht %iith that of X&n sina in the light of are«^ as well as Islamic theories of state in my thesis for the degree of Ph.D.
    [Show full text]
  • Comparison of National Strategies to Promote Artificial Intelligence
    Comparison of National Strategies to Promote Artificial Intelligence Part 2 www.kas.de Imprint Editor: Konrad-Adenauer-Stiftung e. V. 2019, Berlin Authors: Dr. Olaf J. Groth, CEO & Managing Partner Dr. Mark Nitzberg, Principal & Chief Scientist Dan Zehr, Editor-in-Chief Tobias Straube, Project Manager and Senior-Analyst Toni Kaatz-Dubberke, Senior-Analyst Franziska Frische, Analyst Maximilien Meilleur, Analyst Suhail Shersad, Analyst Cambrian LLC, 2381 Eunice Street, Berkeley CA 94708-1644, United States https://cambrian.ai, Twitter: @AICambrian Editorial team and contact at the Konrad-Adenauer-Stiftung e. V. Sebastian Weise Consultant for Global Innovation Policy Post: 10907 Berlin Office: Klingelhöferstraße 23 T +49 30 / 269 96-3516 10785 Berlin F +49 30 / 269 96-3551 [Note: This English-language version was translated from the original publication in German.] Cover image: © sarah5/Mlenny (istockphoto by Getty Images) Images: © p. 11: Matthew Henry, p.16: Manuel Cosentino, p. 23: Dan Gold, p. 30: David Rodrigo, p. 42: Victor Garcia (unsplash); p. 36: byheaven (istockphoto by Getty Images) Design and typesetting: yellow too Pasiek Horntrich GbR The print edition was produced in a carbon-neutral way at Druckerei Kern GmbH, Bexbach, and printed on FSC-certified paper. Printed in Germany. Printed with the financial support of the Federal Republic of Germany. This publication is licenced under the terms of “Creative Commons Attribution-Share Alike 4.0 International”, CC BY-SA 4.0 (available at: https://creativecommons.org/licenses/ by-sa/4.0/) legalcode.de). ISBN 978-3-95721-553-6 Comparison of National Strategies to Promote Artificial Intelligence Part 2 Dr. Olaf J.
    [Show full text]
  • Deep Reinforcement Learning
    DEEP REINFORCEMENT LEARNING Yuxi Li ([email protected]) ABSTRACT We discuss deep reinforcement learning in an overview style. We draw a big pic- ture, filled with details. We discuss six core elements, six important mechanisms, and twelve applications, focusing on contemporary work, and in historical con- texts. We start with background of artificial intelligence, machine learning, deep learning, and reinforcement learning (RL), with resources. Next we discuss RL core elements, including value function, policy, reward, model, exploration vs. exploitation, and representation. Then we discuss important mechanisms for RL, including attention and memory, unsupervised learning, hierarchical RL, multi- agent RL, relational RL, and learning to learn. After that, we discuss RL appli- cations, including games, robotics, natural language processing (NLP), computer vision, finance, business management, healthcare, education, energy, transporta- tion, computer systems, and, science, engineering, and art. Finally we summarize briefly, discuss challenges and opportunities, and close with an epilogue. 1 Keywords: deep reinforcement learning, deep RL, algorithm, architecture, ap- plication, artificial intelligence, machine learning, deep learning, reinforcement learning, value function, policy, reward, model, exploration vs. exploitation, representation, attention, memory, unsupervised learning, hierarchical RL, multi- agent RL, relational RL, learning to learn, games, robotics, computer vision, nat- ural language processing, finance, business management, healthcare, education, energy, transportation, computer systems, science, engineering, art arXiv:1810.06339v1 [cs.LG] 15 Oct 2018 1Work in progress. Under review for Morgan & Claypool: Synthesis Lectures in Artificial Intelligence and Machine Learning. This manuscript is based on our previous deep RL overview (Li, 2017). It benefits from discussions with and comments from many people. Acknowledgements will appear in a later version.
    [Show full text]
  • Can Artificial Intelligence Help Predict a Learner's Needs? Lessons From
    LONDON REVIEW OF EDUCATION e-ISSN: 1474-8479 Journal homepage: https://www.uclpress.co.uk/pages/london-review-of- education Can artificial intelligence help predict a learner’s needs? Lessons from predicting student satisfaction Dimitris Parapadakis How to cite this article Parapadakis, D. (2020) ‘Can artificial intelligence help predict a learner’s needs? Lessons from predicting student satisfaction’. London Review of Education, 18 (2): 178–195. https://doi.org/10.14324/LRE.18.2.03 Submission date: 30 September 2019 Acceptance date: 20 March 2020 Publication date: 21 July 2020 Peer review This article has been peer-reviewed through the journal’s standard double-blind peer review, where both the reviewers and authors are anonymized during review. Copyright © 2020 Parapadakis. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY) 4.0 https://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited. Open access London Review of Education is a peer-reviewed Open Access journal. Parapadakis, D. (2020) ‘Can artificial intelligence help predict a learner’s needs? Lessons from predicting student satisfaction’. London Review of Education, 18 (2): 178–195. DOI https://doi.org/10.14324/LRE.18.2.03 Can artificial intelligence help predict a learner’s needs? Lessons from predicting student satisfaction Dimitris Parapadakis* − University of Westminster, UK Abstract The successes of using artificial intelligence (AI) in analysing large-scale data at a low cost make it an attractive tool for analysing student data to discover models that can inform decision makers in education.
    [Show full text]
  • R'^.'--^^/>^Y^^^,^«»A'*S^-''—^^^^^^^ .^^:•;^::•;A^«Or|L•R•El•Ilt^Ire^,'•'••A; ••.}•..•:J Highest 9F,#In Lotteaiof Pownr.>-U
    .'. •..•.is'.'-im- 'mmmi • .' •.•#^:'•'•r'^.'--^^/>^y^^^,^«»A'*s^-''—^^^^^^^ .^^:•;^::•;A^«or|l•r•el•Ilt^ire^,'•'••A; ••.}•..•:j Highest 9f,#in LotTeaiof Pownr.>-U. S. €ov't Report, Aug. 17,1889^ V'Tbla .luorutitg abiwt.twb o'cioeit; Jobu Wood, tbe well-hnowu plumber ThHr8<l>r» October 18,18$l, \ wbo resides on Main Street, was' awakened by hearing gloss break. Ou IVIieMfleld. " getting out of bed be received ft blow '^\ I Nloo weather for torn hunkliw-AlBledon on tbe bead. He immediately grap* I iHoUool Ulhtrlot. No. S. ara buliaiDK ». now pled with the burglar,: aud after a I brJokBPhool Uouie-CharlMBeemanlibuHd 'Ingoniiddltlon tohUawelllng.-Mrj.Tarrrt,; terrible atruggie succeeded In balding ,of Uratlot county, made her djughtor^JIM. B, hlin until Ills wife obtained help. On I V Kent a vlaltloat week -Mr Oeo Haddle, being searobed at tlie statiou, Mr. IUotKuUiii eouuty, waa. th« guoat of;frjonda lu Wood's watch and wallet were found MASON; MIGH.;™^ thiH vicinity, ferinorlyhla old: home.-Mra. AT CLARK HOUSE, MASON. !: inobortStuadman la atlll very.^lowi chanoea on Ills person. He gave tbe name of ' aro nnulnHt hor rooovery.T-TheJnaeoia are get; Bmder Bobert Terry, Tbe broken glass r tlngiu their, lUnny work at the espenaO:of proved to be a bottle of Sulphur Bitters tlioarly sown wheat.. IfEWS NOTEB. I L. T, Hemaiis is cleaning house nt wblob had almost cured Mrs. Wood of his office. ' "The Aniericdu House Is now vacant, VniTB AHD VINITUBM- rbeutnatisin.—Evohange, 09 See Mr. 'riiorburii's notice of "Man , iillit! AfiSOiJUtEiar PURE Cold ralus the fliat^f the week.
    [Show full text]
  • Deep Reinforcement Learning: an Overview
    DEEP REINFORCEMENT LEARNING:AN OVERVIEW Yuxi Li ([email protected]) ABSTRACT We give an overview of recent exciting achievements of deep reinforcement learn- ing (RL). We discuss six core elements, six important mechanisms, and twelve applications. We start with background of machine learning, deep learning and reinforcement learning. Next we discuss core RL elements, including value func- tion, in particular, Deep Q-Network (DQN), policy, reward, model and planning, exploration, and knowledge. After that, we discuss important mechanisms for RL, including attention and memory, unsupervised learning, transfer learning, multi- agent RL, hierarchical RL, and learning to learn. Then we discuss various appli- cations of RL, including games, in particular, AlphaGo, robotics, natural language processing, including dialogue systems, machine translation, and text generation, computer vision, business management, finance, healthcare, education, Industry 4.0, smart grid, intelligent transportation systems, and computer systems. We mention topics not reviewed yet, and list a collection of RL resources. After pre- senting a brief summary, we close with discussions. This is the first overview about deep reinforcement learning publicly available on- line. It is comprehensive. Comments and criticisms are welcome. (This particular version is incomplete.) Please see Deep Reinforcement Learning, https://arxiv.org/abs/ 1810.06339, for a significant update to this manuscript. arXiv:1701.07274v6 [cs.LG] 26 Nov 2018 1 CONTENTS 1 Introduction5 2 Background7 2.1 Machine Learning . .7 2.2 Deep Learning . .8 2.3 Reinforcement Learning . .9 2.3.1 Problem Setup . .9 2.3.2 Exploration vs Exploitation . .9 2.3.3 Value Function . .9 2.3.4 Dynamic Programming .
    [Show full text]