A project-based guide to undergraduate research in Mathematics starting and sustaining accessible undergraduate research /: starting and sustaining accessible undergraduate research. (2020)

• Record Type: Book
• Further Information: Note: Pamela E. Harris, Erik Insko, Aaron Wootton, editors.
• Other Names: Harris, Pamela E; Insko, Erik; Wootton, Aaron
• Contents: Intro -- Series Preface -- Preface -- Contents -- Folding Words Around Trees: Models Inspired by RNA -- 1 Introduction -- 2 Catalan Numbers -- 3 Valid Plane Trees -- 4 Local Moves and the State Space Graph -- 5 Enumerating Words with Only One Valid Plane Tree -- 6 Enumeration of Valid Plane Trees -- 6.1 Small Values in R(n, 1) -- 6.2 Large Values not in R(n, 1) -- 7 Wobble Pairs and Other Modifications to the Model -- 8 Numerical Models of RNA Folding -- 8.1 Structures of RNA -- 8.2 The RNA Folding Problem -- 8.3 The Inverse RNA Folding Problem -- 9 Conclusion -- References Phylogenetic Networks -- 1 Introduction -- 2 Combinatorics of Phylogenetic Networks -- 2.1 Graphs and Trees -- 2.2 Phylogenetic Networks -- 2.3 Semi-Directed Networks -- 2.4 Restrictions of Networks -- 3 Algebra of Phylogenetic Trees and Networks -- 3.1 Ideals Associated to Trees -- 3.2 Ideals Associated to Sunlet Networks -- 3.3 Beyond Sunlet Networks -- References -- Tropical Geometry -- 1 Tropical Mathematics -- 1.1 Tropical Arithmetic and Tropical Linear Algebra -- 1.2 Tropical Polynomials and Tropical Varieties -- 1.3 Some Tropical Resources -- 2 Tropical Curves in the Plane 2.1 Convex Hulls and Newton Polygons -- 2.2 Subdivisions and the Duality Theorem -- 2.3 The Geometry of Tropical Plane Curves -- 2.4 Skeletons of Tropical Plane Curves -- 3 Tropical Geometry in Three Dimensions -- 3.1 Tropical Surfaces and the Duality Theorem -- 3.2 Tropical Curves in R3 -- 4 Tropicalization -- 4.1 Fields with Valuation -- 4.2 Two Ways to Tropicalize -- 4.3 Tropical Intersections -- References -- Chip-Firing Games and Critical Groups -- 1 Critical Groups -- 1.1 Definitions and Examples -- 1.2 Divisors on a Graph and the Chip-Firing Game -- 1.3 Smith Normal Forms 1.4 Elements of the Critical Group -- 1.5 Spanning Trees and the Matrix Tree Theorem -- 1.6 How Does the Critical Group Change Under Graph Operations? -- 1.7 Which Finite Abelian Groups Occur as the Critical Group of a Graph? -- 1.8 Generators of Critical Groups -- 1.9 Critical Groups of Random Graphs -- 1.10 The Monodromy Pairing on Divisors -- 1.11 Ranks of Divisors and Gonality of Graphs -- 1.12 Chip-Firing on Directed Graphs -- 2 Arithmetical Structures -- 2.1 Definitions and Examples -- 2.2 Counting Arithmetical Structures -- 2.3 Critical Groups of Arithmetical Structures -- References1 Counting Tilings by Taking Walks in a Graph -- 1 Introduction -- 1.1 A First Example -- 1.2 A Second Example -- 1.3 A First Example, Revisited -- 1.3.1 A Second Example, Revisited -- 2 General Approach -- 2.1 Using Linear Algebra Tools -- 2.1.1 Recurrence Relationships and Generating Functions -- 2.1.2 Closed-Form Solutions from Projection -- 3 Using More Entries of the Matrix -- 4 Tiling with Statistics -- 5 Using Other Boards for Tiling -- 5.1 Boards Formed from Triangles -- 5.2 Boards Formed from Hexagons -- 5.3 Three Dimensional Boards -- 6 Using Abstract Tiles -- 6.1 Squaring a Square
• Publisher Details: Cham : Birkhäuser
• Publication Date: 2020
• Extent: 1 online resource (334 p.)
• Subjects: 510.71/1; Mathematics -- Study and teaching (Higher); Mathematics -- Research; Combinatorics & graph theory; Mathematics -- Combinatorics; Mathematics -- Research; Mathematics -- Study and teaching (Higher); Electronic books
• Languages: English
• ISBNs: 9783030378530; 3030378535
• Related ISBNs: 9783030378523
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.