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Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes

1975 Mathematics
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Release : 1975
Genre : Mathematics
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Book Rating : 546/5 ( reviews)

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Book Synopsis Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes by : Thomas Zaslavsky

Download or read book Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes written by Thomas Zaslavsky. This book was released on 1975. Available in PDF, EPUB and Kindle. Book excerpt: An arrangement of hyperplanes of Euclidean or projective d-space is a finite set of hyperplanes, together with the induced partition of the space. Given the hyperplanes of an arrangement, how can the faces of the induced partition be counted? Heretofore this question has been answered for the plane, Euclidean 3-space, hyperplanes in general position, and the d-faces of the hyperplanes through the origin in Euclidean space. In each case the numbers of k-faces depend only on the incidences between intersections of the hyperplane, even though arrangements with the same intersection incidence pattern are not in general combinatorially isomorphic. We generalize this fact by demonstrating formulas for the numbers of k-faces of all Euclidean and projective arrangements, and the numbers of bounded k-faces of the former, as functions of the (semi)lattice of intersections of the hyperplanes, not dependent on the arrangement's combinatorial type.

Facing Up to Arrangements

1975 Automorphic forms
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Release : 1975
Genre : Automorphic forms
Kind : eBook
Book Rating : 541/5 ( reviews)

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Book Synopsis Facing Up to Arrangements by : American Mathematical Society

Download or read book Facing Up to Arrangements written by American Mathematical Society. This book was released on 1975. Available in PDF, EPUB and Kindle. Book excerpt:

Two Papers : H-coextensions of Monoids and the Structure of a Band of Groups

1975 Group extensions (Mathematics)
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Release : 1975
Genre : Group extensions (Mathematics)
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Book Rating : 565/5 ( reviews)

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Book Synopsis Two Papers : H-coextensions of Monoids and the Structure of a Band of Groups by : Jonathan Leech

Download or read book Two Papers : H-coextensions of Monoids and the Structure of a Band of Groups written by Jonathan Leech. This book was released on 1975. Available in PDF, EPUB and Kindle. Book excerpt:

Topics in Hyperplane Arrangements

2017-11-22 Mathematics
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Release : 2017-11-22
Genre : Mathematics
Kind : eBook
Book Rating : 112/5 ( reviews)

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Book Synopsis Topics in Hyperplane Arrangements by : Marcelo Aguiar

Download or read book Topics in Hyperplane Arrangements written by Marcelo Aguiar. This book was released on 2017-11-22. Available in PDF, EPUB and Kindle. Book excerpt: This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.

Arrangements of Hyperplanes

2013-03-09 Mathematics
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Release : 2013-03-09
Genre : Mathematics
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Book Rating : 720/5 ( reviews)

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Book Synopsis Arrangements of Hyperplanes by : Peter Orlik

Download or read book Arrangements of Hyperplanes written by Peter Orlik. This book was released on 2013-03-09. Available in PDF, EPUB and Kindle. Book excerpt: An arrangement of hyperplanes is a finite collection of codimension one affine subspaces in a finite dimensional vector space. Arrangements have emerged independently as important objects in various fields of mathematics such as combinatorics, braids, configuration spaces, representation theory, reflection groups, singularity theory, and in computer science and physics. This book is the first comprehensive study of the subject. It treats arrangements with methods from combinatorics, algebra, algebraic geometry, topology, and group actions. It emphasizes general techniques which illuminate the connections among the different aspects of the subject. Its main purpose is to lay the foundations of the theory. Consequently, it is essentially self-contained and proofs are provided. Nevertheless, there are several new results here. In particular, many theorems that were previously known only for central arrangements are proved here for the first time in completegenerality. The text provides the advanced graduate student entry into a vital and active area of research. The working mathematician will findthe book useful as a source of basic results of the theory, open problems, and a comprehensive bibliography of the subject.

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