Author: Susan Barwick
Publisher: Springer Science & Business Media
ISBN: 038776366X
Category : Mathematics
Languages : en
Pages : 197
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Book Description
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.
Author: Susan Barwick
Publisher: Springer
ISBN: 9780387567648
Category : Mathematics
Languages : en
Pages : 0
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Book Description
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.
Author: Kenneth L. Wantz
Publisher:
ISBN:
Category :
Languages : en
Pages : 196
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Book Description
Author: Andries Evert Brouwer
Publisher:
ISBN:
Category : Block designs
Languages : en
Pages : 18
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Book Description
Author: Frederick W. Stevenson
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 426
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Book Description
Author: Andries Evert Brouwer
Publisher:
ISBN:
Category : Block designs
Languages : en
Pages : 18
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Book Description
Author: Andries Evert Brouwer
Publisher:
ISBN:
Category : Block designs
Languages : en
Pages : 18
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Book Description
Author: Burkard Polster
Publisher: Springer Science & Business Media
ISBN: 1441985263
Category : Mathematics
Languages : en
Pages : 302
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Book Description
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.
Author: Daniel Marshall
Publisher:
ISBN:
Category : Finite fields (Algebra)
Languages : en
Pages : 158
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Book Description
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.
Author: Helmut Salzmann
Publisher: Walter de Gruyter
ISBN: 3110876833
Category : Mathematics
Languages : en
Pages : 705
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Book Description
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)
