| Description |
1 online resource (1 online volumes) : illustrations (black and white) |
| Series |
London Mathematical Society Lecture Note Series ; 470 |
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London Mathematical Society lecture note series ; 470.
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| Contents |
Cover -- Series information -- Title page -- Copyright information -- Contents -- Preface -- The partition complex: an invitation to combinatorial commutative algebra -- 1 Introduction -- 1.1 Overview -- 2 Preliminaries -- 2.1 Simplicial complexes and face rings -- 2.2 Chain complexes -- 2.3 Double complexes -- 3 Cohen-Macaulay Complexes and why we care -- 3.1 The Basic Idea -- 3.2 The Koszul Complex -- 4 The Partition Complex & amp -- Reisner's Theorem -- 4.1 The Partition Complex -- 4.2 Reisner's Theorem -- 5 Partition of Unity -- 5.1 The homology of Tot(P[sup(*)] [otimes] K[sup(*)]) -- 6 Schenzel's Formula -- 7 Poincar[acute(e)] duality -- 7.1 Poincar[acute(e)] duality algebras in general -- 7.2 Poincar[acute(e)] duality for face rings of manifolds -- 7.3 Further remarks on Poincar[acute(e)] duality -- 8 Applications: Triangulations and a conjecture of K[ddot(u)]hnel -- 8.1 The inductive principle and partition of unity -- 8.2 K[ddot(u)]hnel's efficient triangulations of manifolds -- 9 Applications: Subdivisions and the almost-Lefschetz property -- 9.1 Complexes and subdivisions -- 9.2 Partition of unity for disks with induced boundary -- 9.3 The Koszul complex and the Lefschetz property -- 9.4 A modified partition complex -- 9.5 Proof of the subdivision theorem -- Hasse-Weil type theorems and relevant classes of polynomial functions -- 1 Introduction -- 2 Hasse-Weil type theorems -- 3 Preliminaries on algebraic varieties and function fields -- 3.1 Plane curves -- 3.2 Link with function field theory -- 3.3 Equivalence of curves -- 3.4 Kummer or Artin-Schreier covers of curves -- 4 Strategies to investigate the algebraic variety -- 4.1 A method based on intersection multiplicities -- 4.2 Connection with algebraic hypersurfaces -- 5 Permutation polynomials and permutation rational functions |
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5.1 Basic definitions and connections with algebraic curves -- 5.2 Exceptional polynomials and Carlitz Conjecture -- 5.3 A way to decrease the degree of the curve C[sub(f)] -- 5.4 PPs of index [ell] = q + 1 in F[sub(q[sup(2)])] and generalized Niho exponents -- 5.5 Carlitz rank of a permutation polynomial -- 5.6 Complete permutation polynomials and generalization -- 5.7 Permutation rational functions -- 6 Minimal value set polynomials and minimal value set rational functions -- 7 Kloosterman polynomials -- 8 o-polynomials -- 9 Scattered polynomials and maximum scattered linear sets -- 9.1 Exceptional scattered polynomials -- 9.2 Families of non-exceptional scattered polynomials -- 10 Planar functions in odd characteristic -- 10.1 Planar monomials -- 10.2 Planar polynomials -- 10.3 A generalization of planar functions -- 11 Planar functions in even characteristic -- 11.1 Planar monomials -- 11.2 Planar polynomials -- 11.3 Non-exceptional planar polynomials -- 12 APN functions -- 12.1 APN monomials -- 12.2 APN polynomials -- Decomposing the edges of a graph into simpler structures -- 1 Introduction -- 2 What is a discharging argument? -- 3 A toy problem for discharging -- 4 Local arguments in discharging -- 4.1 One reducible configuration, no discharging rule -- 4.2 Two configurations, two discharging rules -- 4.3 Three reducible configurations, two discharging rules -- 4.4 Shifting the density argument to a subgraph of G -- 4.5 Shifting the notion of minimality -- 4.6 Ghost vertices -- 5 Global arguments in discharging -- 5.1 Alternating cycles -- 5.2 Beyond alternating cycles -- 6 Re-colouring -- 6.1 As an intermediary step -- 6.2 As a more general tool -- 7 Conclusion -- Generating graphs randomly -- 1 Introduction -- 2 Preliminaries and Background -- 2.1 Notation and assumptions -- 2.2 Which graph families? -- 2.3 What kind of sampling algorithm? |
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2.4 Sampling graphs with given degrees: an overview -- 3 The configuration model -- 4 Sequential algorithms and graph processes -- 4.1 The regular case -- 4.2 The irregular case, and an almost-FPAUS -- 4.3 Other graph processes -- 5 Switchings-based algorithms -- 5.1 Improvements and extensions -- 6 Markov chain algorithms -- 6.1 Markov chain background -- 6.2 The Jerrum-Sinclair chain -- 6.3 The multicommodity flow method -- 6.4 The switch chain -- 6.6 Restricted graph classes -- 7 Conclusion -- Recent advances on the graph isomorphism problem -- 1 Introduction -- 2 Preliminaries -- 3 The Weisfeiler-Leman Algorithm -- 3.1 The Color Refinement Algorithm -- 3.2 Higher Dimensions -- 3.3 The Power of WL and the WL Dimension -- 3.4 Characterisations of WL -- 4 The Group-Theoretic Graph Isomorphism Machinery -- 4.1 Basics -- 4.2 Luks's Algorithm -- 4.3 Babai's Algorithm -- 4.4 Faster Certificates for Groups with Restricted Composition Factors -- 4.5 From Strings to Hypergraphs -- 5 Quasi-Polynomial Parameterized Algorithms for Isomorphism Testing -- 5.1 Allowing Color Refinement to Split Small Classes -- 5.2 Graphs of Small Genus -- 5.3 Graphs of Small Tree Width -- 5.4 Graphs Excluding Small (Topological) Minors -- 6 Concluding Remarks -- Extremal aspects of graph and hypergraph decomposition problems -- 1 Introduction -- 1.1 Organisation of this survey -- 1.2 Notation -- 2 Approximate and fractional decompositions -- 2.1 From fractional to approximate decompositions -- 2.2 Fractional decomposition thresholds -- 2.3 Bandwidth theorem for approximate decompositions -- 3 Decomposition thresholds for fixed graphs F -- 3.1 Turning approximate decompositions into exact ones -- 3.2 Bipartite graphs -- 3.3 A discretisation result -- 3.4 Decompositions of partite graphs and Latin squares -- 4 F-decompositions of hypergraphs -- 4.1 Minimum degree versions |
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5 Euler tours in hypergraphs -- 5.1 Euler tours: Proof sketch -- 5.2 Open problems on hypergraph decompositions and Euler tours -- 6 Oberwolfach problem -- 6.1 Open problems related to the Oberwolfach problem -- 7 Related decomposition problems -- 7.1 Weighted decompositions into triangles and edges -- 7.2 Packing and covering number -- 7.3 Decomposing highly connected graphs into trees -- 7.4 Tree packings -- 7.5 Sparse decompositions of dense graphs: Erd[H(o)]s meets Nash-Williams -- Borel combinatorics of locally finite graphs -- 1 Introduction -- 2 Notation -- 3 Some Background and Standard Results -- 3.1 Algebras and [sigma]-Algebras -- 3.2 Polish Spaces -- 3.3 The Borel [sigma]-Algebra of a Polish Space -- 3.4 Standard Borel Spaces -- 4 Borel Graphs: Definition and Some Examples -- 5 Basic Properties of Locally Finite Borel Graphs -- 5.1 Borel Colourings -- 5.2 Local Rules -- 5.3 Graphs with a Borel Transversal -- 5.4 Borel Assignments without Small Augmenting Sets -- 6 Negative Results via Borel Determinacy -- 7 Borel Equivalence Relations -- 8 Baire Measurable Combinatorics -- 9 [mu]-Measurable Combinatorics -- 10 Borel Colourings from LOCAL Algorithms -- 11 Borel Results that Use Measures or Baire Category -- 12 Graphs of Subexponential Growth -- 13 Applications to Equidecomposability -- 14 Concluding Remarks -- Codes and designs in Johnson graphs with high symmetry -- 1 Introduction -- 1.1 Codes in Johnson graphs -- 1.2 Designs in Johnson graphs: Delandtsheer designs -- 1.3 Imprimitive examples and a dichotomy -- 1.4 Primitive examples: towards a classi cation -- 2 Completely regular codes and a blow-up Construction -- 3 Sporadic SIT-codes and Delandtsheer 2-designs -- 4 Classical SIT-codes and Delandtsheer designs -- 5 Affine SIT-codes and Delandtsheer designs -- 6 Symplectic SIT-codes and Delandtsheer designs |
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7 Related codes and designs -- 7.1 Transitivity properties on flags and antiflags of designs -- 7.2 SIT-codes and codes in binary Hamming graphs -- Maximal subgroups of nite simple groups: classifications and applications -- 1 Introduction -- 2 The maximal subgroups -- 2.1 Alternating groups -- 2.2 Classical groups -- 2.3 Exceptional groups -- 2.4 Sporadic groups -- 3 Generation and the generating graph -- 4 Base size -- 5 Graph isomorphism and related problems |
| Summary |
These nine articles provide up-to-date surveys of topics of contemporary interest in combinatorics |
| Bibliography |
Includes bibliographical references |
| Notes |
Online resource; title from digital title page (viewed on June 14, 2021) |
| Subject |
Combinatorial analysis -- Congresses
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Análisis combinatorio
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Combinatorial analysis
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Anàlisi combinatòria.
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| Genre/Form |
Conference papers and proceedings
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Congressos.
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Llibres electrònics.
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| Form |
Electronic book
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| Author |
Dabrowski, Konrad K., editor
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| ISBN |
9781009036214 |
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1009036211 |
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