Unitals in Projective Planes

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Publisher : Springer Science & Business Media
ISBN 13 : 038776366X
Total Pages : 197 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Unitals in Projective Planes by : Susan Barwick

Download or read book Unitals in Projective Planes written by Susan Barwick and published by Springer Science & Business Media. This book was released on 2009-04-03 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2,q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds,H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil bound.

Unitals in Projective Planes

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Publisher : Springer
ISBN 13 : 9780387567648
Total Pages : 0 pages
Book Rating : 4.5/5 (676 download)

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Book Synopsis Unitals in Projective Planes by : Susan Barwick

Download or read book Unitals in Projective Planes written by Susan Barwick and published by Springer. This book was released on 2008-11-01 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2,q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds,H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil bound.

Unitals Embedded in Finite Projective Planes

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Publisher :
ISBN 13 :
Total Pages : 196 pages
Book Rating : 4.:/5 (343 download)

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Book Synopsis Unitals Embedded in Finite Projective Planes by : Kenneth L. Wantz

Download or read book Unitals Embedded in Finite Projective Planes written by Kenneth L. Wantz and published by . This book was released on 1995 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Some Unitals on 28 Points and Embeddings in Projective Planes of Order 9

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Publisher :
ISBN 13 :
Total Pages : 18 pages
Book Rating : 4.:/5 (968 download)

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Book Synopsis Some Unitals on 28 Points and Embeddings in Projective Planes of Order 9 by : Andries Evert Brouwer

Download or read book Some Unitals on 28 Points and Embeddings in Projective Planes of Order 9 written by Andries Evert Brouwer and published by . This book was released on 1980 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Projective Planes

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Publisher :
ISBN 13 :
Total Pages : 426 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Projective Planes by : Frederick W. Stevenson

Download or read book Projective Planes written by Frederick W. Stevenson and published by . This book was released on 1992 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Some Unitals on 28 Points and Their Embeddings in Projective Planes of Order 9

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Publisher :
ISBN 13 :
Total Pages : 18 pages
Book Rating : 4.:/5 (959 download)

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Book Synopsis Some Unitals on 28 Points and Their Embeddings in Projective Planes of Order 9 by : Andries Evert Brouwer

Download or read book Some Unitals on 28 Points and Their Embeddings in Projective Planes of Order 9 written by Andries Evert Brouwer and published by . This book was released on 1981 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Some Unitals on 28 Points Ad Their Embeddings in Projective Planes of Order 9

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Publisher :
ISBN 13 :
Total Pages : 18 pages
Book Rating : 4.:/5 (959 download)

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Book Synopsis Some Unitals on 28 Points Ad Their Embeddings in Projective Planes of Order 9 by : Andries Evert Brouwer

Download or read book Some Unitals on 28 Points Ad Their Embeddings in Projective Planes of Order 9 written by Andries Evert Brouwer and published by . This book was released on 1981 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Conics, Unitals and Net Replacement

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Publisher :
ISBN 13 :
Total Pages : 158 pages
Book Rating : 4.:/5 (934 download)

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Book Synopsis Conics, Unitals and Net Replacement by : Daniel Marshall

Download or read book Conics, Unitals and Net Replacement written by Daniel Marshall and published by . This book was released on 2010 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main concerns of this thesis are inherited unitals and conics in finite translation planes. Translation planes may be constructed from particular incidences in other translation planes. One method for doing this is "net-derivation" or the corresponding operation "net replacement". We consider conics and unitals of the finite projective plane PG(2,q2) and observe the effect of net-derivation on their pointsets. Our aim is to determine when the pointsets of conics and unitals of PG(2,q2) are conics and unitals respectively in the translation planes formed after net-derivation. In particular, we focus on t-nest replacement and the corresponding nest-derivation sets. Chapter one introduces all the necessary background on finite affine and projective planes. We consider all relevant substructures and concepts. Of major importance are the definitions of unitals, quadrics, Baer subplanes, Baer sublines and derivation. Chapter two introduces the Bruck-Bose correspondence. We use the Bruck-Bose correspondence extensively in chapters three and four. The Bruck-Bose correspondence is a correspondence between PG(2,q2) and certain incidences in PG(4,q). The key elements are spreads of PG(3,q) as a subspace of PG(4,q). We also detail the known correspondences for Baer sublines, Baer subplanes and unitals as well as the equivalent operation for derivation. Chapter three is where we begin our main work. Here we define net replacement in spreads and show the equivalence to net-derivation sets in PG(2,q2). We look at t-nests in depth, which are an example of net replacement. We prove several known results as well as a host of new geometric and combinatorial properties about t-nests. We show a detailed example of a known t-nest and also define a particular type of replacement set that is common to most t-nests. We finish with examples of different kinds of net-derivation. Chapter four looks at unitals of PG(2,q2) and the effect of general net-derivation. Given a unital of PG(2,q2), suppose we perform net-derivation in PG(2,q2) to form a new translation plane. Can we complete the affine points of the unital to a unital in the new translation plane? We first detail the known results for unitals and derivation. We then prove results for unitals and general net-derivation for all known cases where the net-derivation set lies on l∞. The particular case for t-nests was published separately by the author in [9]. We prove a new result for when the net-derivation set is not on l∞ which is also a new result when just considering derivation. Next, we generalise several other results about derivation and unitals to include general net-derivation. We show the existence of non-inherited unitals in translation planes formed by t-nest replacement of a type that are not present in translation planes formed by derivation. We finish by considering O'Nan configurations contained in unitals in PG(2,q2) and planes formed by net-derivation. Chapter five considers conics and the effect of multiple derivation. Given a conic of PG(2,q2), suppose we perform net-derivation in PG(2,q2) to form a new translation plane. Can we complete the affine points of the conic to a conic in the new translation plane? In particular, we focus on inherited conics with respect to multiple derivation. We begin by defining notation and present a new corollary on nest-derivation and conics, followed by several basic theorems on conics and derivation. We then present, in three stages, a novel characterisation of the equations of conics that are not arcs after derivation with the real derivation set. Next we provide a brief survey of the known results for inherited conics and derivation. We then restrict our attention to conics contained in a particular family Ccd. Using this family, we prove several new theorems on the existence of inherited (q2+1)-arcs in a class of planes formed by double derivation in PG(2,q2), where q is odd. We follow this by computing an example of a complete 24-arc in a particular translation plane of order 25. Finally, we show the existence of a family of inherited arcs in a class of Andre planes which includes the regular Nearfield planes of odd order.

The Automorphism Groups of Unitary Block Designs and the Existence of O'Nan Configurations

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Author :
Publisher : Open Dissertation Press
ISBN 13 : 9781361382097
Total Pages : pages
Book Rating : 4.3/5 (82 download)

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Book Synopsis The Automorphism Groups of Unitary Block Designs and the Existence of O'Nan Configurations by : Yee-Ka Tai

Download or read book The Automorphism Groups of Unitary Block Designs and the Existence of O'Nan Configurations written by Yee-Ka Tai and published by Open Dissertation Press. This book was released on 2017-01-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "The Automorphism Groups of Unitary Block Designs and the Existence of O'Nan Configurations" by Yee-ka, Tai, 戴怡嘉, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: A unital is a 2-(n DEGREES3 + 1, n + 1, 1) design. An important invariant of a unital is its automorphism group. A projective plane is a 2-(m DEGREES2 +1, m+1, 1) design. A polar unital is a unital that consists of the absolute points and non-absolute lines of a unitary polarity in a projective plane. The collineation group stabilizing a polar unital in a projective plane is always a subgroup of the automorphism group of the unital. The classical example of a unital is a classical unital H of order q. It is an embedded unital in PG(2, q DEGREES2). In 1972 O'Nan [O'Nan, 1972] proved that Aut(H), the design automorphism group, is isomorphic to Col(H), the collineation subgroup of PG(2, q DEGREES2) stabilizing H. It was also observed in [O'Nan, 1972] that in the classical unital there is no O'Nan configuration: four (non-absolute) lines intersecting in six (absolute) points. In 1981 Piper [Piper, 1981] conjectured that the non-existence of O'Nan configurations characterizes the classical unital. In this thesis, we study two classes of unitary block designs: the Ganley unital in the Dickson semifield plane and the Figueroa unital in the Figueroa plane. We study the existence of O'Nan configurations in these unitals and also investigate their automorphism groups. A Ganley unital is defined by a unitary polarity in a Dickson semifield plane [Ganley, 1972]. For a Dickson semifield plane II(K(σ)), let U(σ) be the Ganley unital defined. We prove that every Ganley unital U(σ), parametrized by a field automorphism σ, is non-classical, extending a result of Ganley's [Ganley, 1972]; we prove that U(σ_1) is isomorphic to U(σ_2) if and only if σ_1 = σ_2 or σ_1 = σ_2 DEGREES(-1); and we determine the automorphism group of U(σ). The finite Figueroa plane is another class of non-Desarguesian projective planes [Figueroa, 1982; Hering and Schaeffer, 1982]. A synthetic construction of the finite Figueroa plane is known [Grundhofer, 1986]. A Figueroa planes of finite square order possess a unitary polarity [de Resmini and Hamilton, 1998]. The unital defined is called the Figueroa unital. We introduce an alternative synthetic description of the Figueroa plane leading to an alternative synthetic description of the Figueroa unital. We demonstrate the existence of O'Nan configurations in the Figueroa unital, thus providing support to Piper's conjecture. We extend and complete some of the partial structural results obtained in [Hui and Wong, 2012], and also demonstrate the existence of many classical unitals of order q in a Figueroa unital of order q DEGREES3 DOI: 10.5353/th_b5295508 Subjects: Automorphisms Projecti

Handbook of Finite Translation Planes

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Publisher : CRC Press
ISBN 13 : 1420011146
Total Pages : 884 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Handbook of Finite Translation Planes by : Norman Johnson

Download or read book Handbook of Finite Translation Planes written by Norman Johnson and published by CRC Press. This book was released on 2007-02-15 with total page 884 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook of Finite Translation Planes provides a comprehensive listing of all translation planes derived from a fundamental construction technique, an explanation of the classes of translation planes using both descriptions and construction methods, and thorough sketches of the major relevant theorems. From the methods of Andre to coordi

Handbook of Combinatorial Designs

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Publisher : CRC Press
ISBN 13 : 1420010549
Total Pages : 1011 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Handbook of Combinatorial Designs by : Charles J. Colbourn

Download or read book Handbook of Combinatorial Designs written by Charles J. Colbourn and published by CRC Press. This book was released on 2006-11-02 with total page 1011 pages. Available in PDF, EPUB and Kindle. Book excerpt: Continuing in the bestselling, informative tradition of the first edition, the Handbook of Combinatorial Designs, Second Edition remains the only resource to contain all of the most important results and tables in the field of combinatorial design. This handbook covers the constructions, properties, and applications of designs as well as existence

Geometries and Groups

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Publisher : Springer
ISBN 13 : 3540386394
Total Pages : 262 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Geometries and Groups by : M. Aigner

Download or read book Geometries and Groups written by M. Aigner and published by Springer. This book was released on 2006-11-14 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dedicated to Professor Dr. Hanfried Lenz on the Occasion of his 65th Birthday

Topics on Steiner Systems

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Publisher : Elsevier
ISBN 13 : 0080867707
Total Pages : 361 pages
Book Rating : 4.0/5 (88 download)

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Book Synopsis Topics on Steiner Systems by :

Download or read book Topics on Steiner Systems written by and published by Elsevier. This book was released on 2011-08-26 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topics on Steiner Systems

Handbook of Finite Fields

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Publisher : CRC Press
ISBN 13 : 1439873828
Total Pages : 1048 pages
Book Rating : 4.4/5 (398 download)

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Book Synopsis Handbook of Finite Fields by : Gary L. Mullen

Download or read book Handbook of Finite Fields written by Gary L. Mullen and published by CRC Press. This book was released on 2013-06-17 with total page 1048 pages. Available in PDF, EPUB and Kindle. Book excerpt: Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and

Compact Projective Planes

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Publisher : Walter de Gruyter
ISBN 13 : 3110876833
Total Pages : 705 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Compact Projective Planes by : Helmut Salzmann

Download or read book Compact Projective Planes written by Helmut Salzmann and published by Walter de Gruyter. This book was released on 2011-06-24 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Designs and Their Codes

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Publisher : Cambridge University Press
ISBN 13 : 9780521458399
Total Pages : 366 pages
Book Rating : 4.4/5 (583 download)

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Book Synopsis Designs and Their Codes by : E. F. Assmus

Download or read book Designs and Their Codes written by E. F. Assmus and published by Cambridge University Press. This book was released on 1994-01-06 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: A self-contained account suited for a wide audience describing coding theory, combinatorial designs and their relations.

A Geometrical Picture Book

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Publisher : Springer Science & Business Media
ISBN 13 : 1441985263
Total Pages : 302 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis A Geometrical Picture Book by : Burkard Polster

Download or read book A Geometrical Picture Book written by Burkard Polster and published by Springer Science & Business Media. This book was released on 2012-09-17 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: How do you convey to your students, colleagues and friends some of the beauty of the kind of mathematics you are obsessed with? If you are a mathematician interested in finite or topological geometry and combinatorial designs, you could start by showing them some of the (400+) pictures in the "picture book". Pictures are what this book is all about; original pictures of everybody's favorite geometries such as configurations, projective planes and spaces, circle planes, generalized polygons, mathematical biplanes and other designs which capture much of the beauty, construction principles, particularities, substructures and interconnections of these geometries. The level of the text is suitable for advanced undergraduates and graduate students. Even if you are a mathematician who just wants some interesting reading you will enjoy the author's very original and comprehensive guided tour of small finite geometries and geometries on surfaces This guided tour includes lots of sterograms of the spatial models, games and puzzles and instructions on how to construct your own pictures and build some of the spatial models yourself.