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The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices

2014 Geometry, Algebraic
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Release : 2014
Genre : Geometry, Algebraic
Kind : eBook
Book Rating : 922/5 ( reviews)

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Book Synopsis The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices by : Peter Šemrl

Download or read book The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices written by Peter Šemrl. This book was released on 2014. Available in PDF, EPUB and Kindle. Book excerpt: "November 2014, volume 232, number 1089 (first of 6 numbers)"

The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices

2014-09-29 Mathematics
Download The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices PDF Online Free

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Release : 2014-09-29
Genre : Mathematics
Kind : eBook
Book Rating : 450/5 ( reviews)

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Book Synopsis The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices by : Peter Šemrl

Download or read book The Optimal Version of Hua's Fundamental Theorem of Geometry of Rectangular Matrices written by Peter Šemrl. This book was released on 2014-09-29. Available in PDF, EPUB and Kindle. Book excerpt: Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all m\times n matrices over a division ring \mathbb{D} which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the non-square case.

Geometry Of Semilinear Embeddings: Relations To Graphs And Codes

2015-05-28 Mathematics
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Release : 2015-05-28
Genre : Mathematics
Kind : eBook
Book Rating : 095/5 ( reviews)

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Book Synopsis Geometry Of Semilinear Embeddings: Relations To Graphs And Codes by : Mark Pankov

Download or read book Geometry Of Semilinear Embeddings: Relations To Graphs And Codes written by Mark Pankov. This book was released on 2015-05-28. Available in PDF, EPUB and Kindle. Book excerpt: This volume covers semilinear embeddings of vector spaces over division rings and the associated mappings of Grassmannians. In contrast to classical books, we consider a more general class of semilinear mappings and show that this class is important. A large portion of the material will be formulated in terms of graph theory, that is, Grassmann graphs, graph embeddings, and isometric embeddings. In addition, some relations to linear codes will be described. Graduate students and researchers will find this volume to be self-contained with many examples.

Brandt Matrices and Theta Series over Global Function Fields

2015-08-21 Mathematics
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Release : 2015-08-21
Genre : Mathematics
Kind : eBook
Book Rating : 198/5 ( reviews)

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Book Synopsis Brandt Matrices and Theta Series over Global Function Fields by : Chih-Yun Chuang

Download or read book Brandt Matrices and Theta Series over Global Function Fields written by Chih-Yun Chuang. This book was released on 2015-08-21. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this article is to give a complete account of the Eichler-Brandt theory over function fields and the basis problem for Drinfeld type automorphic forms. Given arbitrary function field k together with a fixed place ∞, the authors construct a family of theta series from the norm forms of "definite" quaternion algebras, and establish an explicit Hecke-module homomorphism from the Picard group of an associated definite Shimura curve to a space of Drinfeld type automorphic forms. The "compatibility" of these homomorphisms with different square-free levels is also examined. These Hecke-equivariant maps lead to a nice description of the subspace generated by the authors' theta series, and thereby contributes to the so-called basis problem. Restricting the norm forms to pure quaternions, the authors obtain another family of theta series which are automorphic functions on the metaplectic group, and this results in a Shintani-type correspondence between Drinfeld type forms and metaplectic forms.

A Geometric Theory for Hypergraph Matching

2014-12-20 Mathematics
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Release : 2014-12-20
Genre : Mathematics
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Book Rating : 658/5 ( reviews)

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Book Synopsis A Geometric Theory for Hypergraph Matching by : Peter Keevash

Download or read book A Geometric Theory for Hypergraph Matching written by Peter Keevash. This book was released on 2014-12-20. Available in PDF, EPUB and Kindle. Book excerpt: The authors develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, they introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. They determine the essentially best possible minimum degree sequence for finding an almost perfect matching. Furthermore, their main result establishes the stability property: under the same degree assumption, if there is no perfect matching then there must be a space or divisibility barrier. This allows the use of the stability method in proving exact results. Besides recovering previous results, the authors apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in -graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemerédi Theorem. Here they prove the exact result for tetrahedra and the asymptotic result for Fischer's conjecture; since the exact result for the latter is technical they defer it to a subsequent paper.

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