Custom cover image
Algorithmic Algebraic Combinatorics and Gröbner Bases / edited by Mikhail Klin, Gareth A. Jones, Aleksandar Jurišić, Mikhail Muzychuk, Ilia Ponomarenko
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2009Description: Online-Ressource (XII, 311p, digital)ISBN:- 9783642019609
- 1282827022
- 9783642019593
- 9781282827028
- 511.6
- QA164-167.2
- QA164
Contents:
Summary: Tutorials -- Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics -- Siamese Combinatorial Objects via Computer Algebra Experimentation -- Using Gröbner Bases to Investigate Flag Algebras and Association Scheme Fusion -- Enumerating Set Orbits -- The 2-dimensional Jacobian Conjecture: A Computational Approach -- Research Papers -- Some Meeting Points of Gröbner Bases and Combinatorics -- A Construction of Isomorphism Classes of Oriented Matroids -- Algorithmic Approach to Non-symmetric 3-class Association Schemes -- Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings -- A Construction of Designs from PSL(2,q) and PGL(2,q), q=1 mod 6, on q+2 Points -- Approaching Some Problems in Finite Geometry Through Algebraic Geometry -- Computer Aided Investigation of Total Graph Coherent Configurations for Two Infinite Families of Classical Strongly Regular Graphs.Summary: This collection of tutorial and research papers introduces readers to diverse areas of modern pure and applied algebraic combinatorics and finite geometries with a special emphasis on algorithmic aspects and the use of the theory of Gröbner bases. Topics covered include coherent configurations, association schemes, permutation groups, Latin squares, the Jacobian conjecture, mathematical chemistry, extremal combinatorics, coding theory, designs, etc. Special attention is paid to the description of innovative practical algorithms and their implementation in software packages such as GAP and MAGMA. Readers will benefit from the exceptional combination of instructive training goals with the presentation of significant new scientific results of an interdisciplinary nature.PPN: PPN: 1648808476Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Preface; Contents; Contributors; Part A Tutorials; Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics; Introduction; Preliminaries; Main Notions; Classification; Regular Subgroups; Principal Loop-Isotopes of Quasigroups; Classics and Folklore; General Links Between Groups; Geometrical and Nongeometrical SRG's; Factorization of Latin Square Graphs; The Group Case; Main Class of a Group Case; Garden of Small Examples; Main Results; The Case n=2; The Case n=3; The Case n=4; The Case n=5, Part a; The Case n=5, Part b; The Case n=6
The Remark of Barlotti and StrambachComputer Aided Answer; Computer-based Analysis of the Example with GAP; Further Computer-based Analysis with COCO; Computer Free Interpretation; General Idea; Axioms of TD(3,6); Description of a Model; Fulfillment of Axioms; The Group Aut(S); Transversal Designs of Groups of Order 6; Regular Subgroup for the Loop Case; Exceptional Quasigroup Re-visited; More Examples; Some Preliminary Observations; Defining Points and Lines; Constructing a Transversal Design; Automorphisms of the Design; Analyzing the Results; Conclusion; Partial Difference Sets and S-rings
Pseudo-geometrical GraphsThe Group Case; Hamming Graphs and Latin Squares; Loops and Permutations; Order 5; Order 6; Vertex-transitive Graphs; Q6; Q2p; Erich Schönhardt; Article by Brendan McKay et al.; A Few Books; More References; The Presented Project; Acknowledgments; References; Siamese Combinatorial Objects via Computer Algebra Experimentation; Introduction; Preliminaries; Color Graphs; Coherent Configurations and Association Schemes; Incidence Structures; General Definitions; Steiner Systems; Generalized Quadrangles; Kramer-Mesner Method and Related Issues; Double Cosets
Computer Algebra ToolsComputations in Combinatorics; COCO; GAP; GRAPE; nauty; Computer Algebra Experimentation; Explanation Versus Interpretation; Siamese Objects: Main Definitions; Siamese Color Graphs; Siamese Association Schemes; Siamese Steiner Designs; Pattern of Investigation; Siamese Graphs as Simultaneous Antipodal Covers; Review of Main Results; Initial Example on 15 Points; Data from COCO; Theoretical Interpretation; A Few Words About STS(15); Automorphism Group of a Siamese Partition for STS(15)#1; Summary of Known Results; Other Roads to Group N; Subdirect Products
Faithful Actions of N on 15 PointsExplicit Desired Action of N on 15 Points; Analytic Enumeration of Orbits of (N,Omega{}3); Constructive Enumeration of Orbits of (N,Omega{}3); Summary of Results About N; More About 15 Points; Starting Group; A Non-coherent Siamese Partition of STS(15)#7; Description of the Model of STS(15)#7; All Siamese Color Graphs on 15 Points are Obtained; Objects on 40 Points; Classical Objects; Circulant Example; One More Point-Transitive Example; Other Siamese Objects; Discussion on Methodology; Strategy A: Combinatorial Analogue of Transitive Extension
Strategy B: Construction and Investigation of Steiner Designs
No physical items for this record