Advances in the Mathematical Sciences: AWM Research Symposium, Los Angeles, CA, April 2017 (Association for Women in Mathematics Series, 15)

"Featuring research from the 2017 research symposium of the Association for Women in Mathematics, this volume presents recent findings in pure mathematics and a range of advances and novel applications in fields such as engineering, biology, and medicine. Featured topics include geometric group theory, generalized iterated wreath products of cyclic groups and symmetric groups, Conway-Coxeter friezes and mutation, and classroom experiments in teaching collegiate mathematics. A review of DNA topology and a computational study of learning-induced sequence reactivation during sharp-wave ripples are also included in this volume. Numerous illustrations and tables convey key results throughout the book. This volume highlights research from women working in academia, industry, and government. It is a helpful resource for researchers and graduate students interested in an overview of the latest research in mathematics"-- Provided by publisher About the 2017 AWM Research Symposium......Page 7 About This Volume......Page 8 Acknowledgments......Page 11 Contents......Page 12 1 Introduction......Page 14 2 Geodesic Metric Spaces, Isometries and Quasi-Isometries......Page 15 3 Hyperbolic Groups......Page 19 4 Beyond Hyperbolicity......Page 22 References......Page 27 1 Introduction......Page 28 2.1 Wreath Products......Page 31 2.2 Clifford Theory......Page 32 2.3 Rooted Trees of a Fixed Height......Page 33 2.4 Bratteli Diagrams......Page 34 3.1 Number of Irreducible Representations......Page 35 4 Bijection Between the Branching Diagram for Generalized Iterated Wreath Products and Rooted Trees......Page 36 4.1 Degrees of Irreducible Representations......Page 37 5 Fast Fourier Transforms, Adapted Bases, and Upper Bound Estimates......Page 38 References......Page 40 1 Introduction......Page 42 2 Background......Page 45 2.2 Wreath Products......Page 46 2.3 Clifford Theory......Page 48 2.4 Rooted Trees of a Fixed Height......Page 49 2.5 Bratteli Diagrams......Page 51 2.6 Adapted Bases and Fast Fourier Transforms......Page 52 3 Irreducible Representations of Iterated Wreath Products......Page 53 4.1 Degrees of Irreducible Representations......Page 55 5 Fast Fourier Transforms, Adapted Bases, and Upper Bound Estimates......Page 56 References......Page 57 Conway–Coxeter Friezes and Mutation: A Survey......Page 60 1 Introduction......Page 61 1.1 Frieze Patterns......Page 62 1.2 Cluster Algebras......Page 63 1.3 Cluster Categories......Page 64 2 From Cluster Categories to Frieze Patterns......Page 65 3 Description of the Regions in the Frieze......Page 68 3.2 Diagonal Defines Two Rays......Page 69 3.3 Diagonal Defines Subsets of Indecomposables......Page 70 4 Mutating Friezes......Page 73 4.2 The Effect of Flips on Friezes......Page 75 4.3 Mutation of Regions......Page 76 4.4 Mutation of Frieze......Page 77 References......Page 80 1 Introduction......Page 82 2 Preliminaries......Page 83 3 Decomposition of Q......Page 86 4 H-orbits of Qu......Page 88 References......Page 90 1 Introduction and Statement of Results......Page 91 2.1 The Small Period Matrix of the Jacobian of a Curve......Page 93 2.2 The Two-Torsion on the Jacobian of a Hyperelliptic Curve......Page 96 2.3 Mumford and Poor's Vanishing Theorem......Page 98 3.1 Counting the Allowable Sets U......Page 100 3.2 Counting the Different Sets U(Ω,m) for a Hyperelliptic Curve......Page 102 References......Page 107 1 Introduction......Page 108 2 Categorical Context, Cross Effects, Linearization,and Directional Derivatives......Page 109 2.1 Categorical Context......Page 110 2.2 Cross Effects and Linearization......Page 111 2.3 Higher Order Directional Derivatives......Page 113 3 The Second-Order Directional Derivative of a Composition......Page 116 4 A Composition of Directional Derivatives......Page 121 5 Proof of the Chain Rule for the Second Directional Derivative......Page 126 6 Conclusion......Page 129 References......Page 130 1 Introduction......Page 131 2 Knots and Links......Page 132 3 Tangles......Page 137 4 Biology Background......Page 140 4.1 Transcription and Replication......Page 142 4.2 Topoisomerase......Page 145 4.3 Recombinase......Page 146 5 Tangle Model......Page 148 6 Further Applications of the Tangle Model......Page 151 References......Page 152 Structural Identifiability Analysis of a Labeled Oral Minimal Model for Quantifying Hepatic Insulin Resistance......Page 155 1 Introduction......Page 156 2 Model Derivations......Page 157 2.1 Derivation of Oral Minimal Model (OMM)......Page 158 2.2 Derivation of Labeled Oral Minimal Model (OMM*)......Page 160 3 Structural Identifiability Analysis of OMM*......Page 161 3.2 Determining Structural Identifiability of OMM*......Page 162 4 Discussion......Page 167 References......Page 169 Spike-Field Coherence and Firing Rate Profiles of CA1 Interneurons During an Associative Memory Task......Page 171 1 Introduction......Page 172 2.1 Behavioral Paradigm......Page 173 2.3 Local Field Potential and Spike-Phase Coherence Analyses......Page 174 3 Results......Page 175 4 Discussion......Page 178 References......Page 180 Learning-Induced Sequence Reactivation During Sharp-Wave Ripples: A Computational Study......Page 182 1 Introduction......Page 183 2.1 Changes in the Minimal Number of Synapses can Promote Reactivation of Cell Triplet in CA3......Page 185 2.2 Pre-Sleep Activation Modulates Learning-Induced Gain in Post-Sleep Reactivation......Page 189 2.3 Simplified Learning Extends Word Length Reactivation......Page 191 2.4 Experience-Related Learning: From Rat Trajectory to New Synaptic Connections......Page 195 2.5 Experience Learning Increases Trajectory Reactivation in Post-Sleep......Page 198 3 Conclusion......Page 201 Sleep Model Representation of Sleep Activity......Page 203 Rationale......Page 204 Equations and Parameters......Page 205 Connectivity......Page 206 Introducing Targeted Synaptic Changes......Page 207 4.2 Modeling a Virtual Rat's Spatial Learning Experience......Page 209 References......Page 210 1 Introduction......Page 214 2 Mathematical Model......Page 215 2.1 Mass Conservation Laws......Page 216 2.2 The Isothermal Two-Phase Two-Component Model......Page 219 3.1 Standard DG Discretization......Page 220 3.2 The Partial Upwind Method......Page 221 3.3 General PDE Model......Page 222 3.4 Numerical Discretization......Page 223 3.5 The Newton–Raphson Method for Linearization......Page 226 4.1 CO2 Injection Test on Smooth Solutions......Page 227 Example of Homogeneous Medium......Page 229 Example of Heterogenous Medium......Page 232 CO2 Injection Simulation for Different Orders of Approximation......Page 233 4.4 Injection of CO2 into Heterogeneous Porous Medium......Page 238 References......Page 240 1 Introduction......Page 242 2 Background......Page 244 3 Continuous Dependence on Modeling......Page 245 4 Regularization......Page 250 References......Page 252 Research in Collegiate Mathematics Education......Page 254 1 Introduction......Page 255 2.1 Methods......Page 256 2.2 Results......Page 257 3 Gesture in Teaching Calculus......Page 258 3.2 Results......Page 259 4 Combinatorics......Page 261 4.2 Results......Page 262 5.1 Methods: IOLA Materials......Page 264 5.2 Results: Illustrative Example......Page 265 5.3 Discussion......Page 267 6.1 Methods......Page 268 6.2 Results......Page 269 7 Mathematics Applied to Teaching......Page 271 7.1 Methods: Motivation......Page 272 7.3 Discussion......Page 273 References......Page 274 Index......Page 278