[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116696-en":3,"doc-seo-116696-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},116696,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Geometry of Derivation - Volume II: Theory of Skewfield Flocks","Geometry of Derivation - Volume II focuses on the theory of skewfield flocks and the connections to derivable nets and translation planes in both finite and infinite settings. The work develops the required theoretical framework to construct concrete examples and support research in the area. It explains how to extend affine-plane derivation ideas through conditions for the existence of spreads, using determinant constructions and unwrapping principles, while also providing detailed background from non-commutative algebra for a geometric treatment.","Geometry of Derivation  \nThis book is concerned mainly with the theory offlocks over skewfields. It begins with discussing what conditions would be required to find a possible way to extend flocks of hyperbolic quadrics and flocks of quadratic cones. This theory completely changes the idea of derivation of an affine plane that contains a derivable net.  \nThis volume will give the necessary theory for the reader to understand how to construct examples and become researchers in the field. It shows how to construct four types of determinants, the (i,j)-determinants, which if never zero for the non-zero matrices of the spread will indicate that the first condition for existence of a spread then holds. If applicable, the left unwrapping principle, if this also is valid, will show that a left flock spread is constructed.  \nThe book continues the presentation in Geometry of Derivation with Applications, Volume I, and a third volume, Geometry of Derivation, Volume III: Classification of Skewfield Flocks (2026) is also available, both from CRC Press. This is the sixth work in a longstanding series of books on combinatorial geometry by the author, including Subplane Covered Nets, Johnson (2000); Foundations of Translation Planes, Biliotti, Jha, and Johnson (2001); Handbook of Finite Translation Planes, Johnson, Jha, and Biliotti (2007); and Combinatorics of Spreads and Parallelisms, Johnson (2010), all published by CRC Press.  \nLike its predecessors, this book will primarily deal with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. Translation planes over non-commutative skewfields have not traditionally hada significant representation in incidence geometry, and derivable nets over skewfields have only been marginally understood. Both are deeply examined in this volume, while ideas of non-commutative algebra are also described in detail, with all the necessary background given a geometric treatment.  \nGeometry of Derivation  \nVolume II: Theory of Skewfield Flocks  \nNorman L. Johnson  \nDesigned cover image: Norman L. Johnson and Bonnie L. Hemenover  \nFirst edition published 2026 by CRC Press  \n2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431  \nand by CRC Press  \n4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RNCRC Press is an imprint of Taylor & Francis Group, LLC © 2026 Norman L. Johnson  \nReasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproducedin this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.  \nExcept as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.  \nFor permission to photocopy or use material electronically from this work, access [www.copyright](www.copyright). com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermis[sions@tandf.co.uk](sions@tandf.co.uk)  \nFor Product Safety Concerns and Information please contact our EU representative GPSR@tay[lorandfrancis.com. Taylor & Francis Verlag GmbH](lorandfrancis.com. Taylor & Francis Verlag GmbH), Kaufingerstraße 24, 80331 München, Germany.  \nTrademark notice: Product or corporate names maybe trademarks or registered trademarks and are used only for identification ","cbCaihoNtqStuuJW","https://ap.wps.com/l/cbCaihoNtqStuuJW","pdf",18560636,1,341,"English","en",105,"# Part 1. When quasifibrations become spreads\n## Chapter 1. Quasifibrations\n## Chapter 2. Unwrapping\n## Chapter 3. Twisted extensions\n## Chapter 4. Semifield planes from cyclic algebras\n## Chapter 5. Proper quasifibrations of dimension 2\n# Part 2. Skewfield flocks – A window\n## Chapter 6. The main points and ideas\n# Part 3. Foundations of flock theory\n## Chapter 7. Building the foundation\n## Chapter 8. Generic and non-generic flocks\n# Part 4 . Framework for flock theory\n## Chapter 9 . Setting up flock and spread connections\n# Part 5 . Left σ−A flocks and spreads\n## Chapter 10 . Left σ − A – hyperbolic flocks\n## Chapter 11 . Left σ − A – conical flocks; 1st and 2nd main theorems\n## Chapter 12 . The lower left form theory\n## Chapter 13 . The 1st general theorem of flocks over skewfields","[{\"question\":\"What topic does Volume II focus on?\",\"answer\":\"Volume II develops the theory of skewfield flocks and how it connects to derivable nets and translation planes in finite and infinite cases.\"},{\"question\":\"How does the volume support constructing examples?\",\"answer\":\"It provides the necessary theoretical framework and criteria for existence of spreads, including determinant-based constructions and unwrapping principles, enabling explicit constructions.\"},{\"question\":\"What algebraic background is included?\",\"answer\":\"The book describes key ideas from non-commutative algebra in detail and supplies the needed background for a geometric treatment of flock theory.\"}]","Geometry of Derivation - Volume II: Theory of Skewfield Flocks | PDF",1785662596,859,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"geometry-of-derivation-volume-ii-theory-of-skewfield-flocks","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/geometry-of-derivation-volume-ii-theory-of-skewfield-flocks/116696/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What topic does Volume II focus on?","Question",{"text":75,"@type":76},"Volume II develops the theory of skewfield flocks and how it connects to derivable nets and translation planes in finite and infinite cases.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the volume support constructing examples?",{"text":80,"@type":76},"It provides the necessary theoretical framework and criteria for existence of spreads, including determinant-based constructions and unwrapping principles, enabling explicit constructions.",{"name":82,"@type":73,"acceptedAnswer":83},"What algebraic background is included?",{"text":84,"@type":76},"The book describes key ideas from non-commutative algebra in detail and supplies the needed background for a geometric treatment of flock theory.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]