Advances in Mathematical Sciences: AWM Research Symposium, Houston, TX, April 2019 (Association for Women in Mathematics Series (21))

Preface
	About the 2019 AWM Research Symposium
	About This Volume
	Acknowledgments
Contents
Part I From the Plenary Talks
	Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints (Survey)
		1 Model Problem
		2 Finite Element Methods
			2.1 Conforming Methods
			2.2 Classical Nonconforming Methods
			2.3 Discontinuous Galerkin Methods
			2.4 Mixed Methods
			2.5 Finite Element Methods for the Optimal Control Problem
		3 Convergence Analysis
		4 Concluding Remarks
		References
Part II Algebraic Combinatorics and Graph Theory
	Some q-Exponential Formulas Involving the Double Lowering Operator ψ for a Tridiagonal Pair (Research)
		1 Introduction
		2 Preliminaries
		3 The First and Second Split Decomposition of V
		4 The Linear Transformation ψ
		5 The Linear Transformation
		6 The Linear Transformation M
		7 A Factorization of
		8 The Eigenvalues and Eigenspaces of M
		9 The Actions of ψ, K,B,,A,A on {Wi}i=0d
		10 The Actions of M1 on {Ui}i=0d,{Ui"322B37F }i=0d, {EiV}i=0d, {EiV}i=0d
		11 When  Is a Leonard System
		References
	Distance Graphs Generated by Five Primes (Research)
		1 Introduction
		2 Three Lemmas on κ(D)
		3 Class 3 Prime Sets of the Form {2,3,7,p,q}
		4 Class 4 Prime Sets of the Form {2,3,7,19,p}
			4.1 Characterizing Gap Sequences
			4.2 Characterizing Color Sequences
			4.3 Guaranteed Partial Sums
		5 Conclusion
		References
	Combinatorial Characterization of Queer Supercrystals (Survey)
		1 Introduction
		2 Queer Supercrystals
		3 Local Axioms
		4 Graph on Type A Components
		5 Characterization of Queer Crystals
		References
	Enumerating in Coxeter Groups (Survey)
		1 Introduction
		2 Main Tools
		3 Counting Special Elements: An Example
		4 Counting an Element's Special Features: An Example
		5 Directions for Future Research
		References
Part III Algebraic Biology
	From Chaos to Permanence Using Control Theory (Research)
		1 Introduction
		2 Survival and Chaos
		3 The Chaotic Control Algorithm (CCA)
		4 Applications of the CCA to Predator-Prey Models
			4.1 Control Through Harvesting in a Predator-Prey Model
				4.1.1 Boundedness of the Solutions
				4.1.2 The Harvesting Model
				4.1.3 Application of CCA SKCP
			4.2 Control Through Harvesting in a Food-Chain System
				4.2.1 Equilibrium Analysis SKCP
				4.2.2 Conditions for Permanence
				4.2.3 Control Algorithm Using Harvesting SKCP
		5 Conclusions
		6 MATLAB Code to Determine Permanence Using Boundary Rest Points SKCP
		References
	Classification on Large Networks: A Quantitative Bound via Motifs and Graphons (Research)
		1 Introduction
			1.1 Contributions
			1.2 Related Literature
			1.3 Organization
		2 Background
			2.1 Decorated graphons
			2.2 Spectra and Wasserstein Distances
		3 Graphons for Classification: Main Results
		4 An Application: Classification of Lupus Erythematosus
		5 Conclusion
		6 Proof of Theorem 1
			6.1 Auxiliary Results
			6.2 Proof of Theorem 1
		7 Proof of Theorem 2
			7.1 Auxiliary Results
			7.2 Proof of Theorem 2
		8 A Similar Bound for Degree Features
		9 Heterogenous Sample Sizes
		References
	Gröbner Bases of Convex Neural Code Ideals (Research)
		1 Introduction and Background
		2 Convexity of Neural Codes
		3 Structure of the Neural Ideal
			3.1 Gröbner Basis of a Neural Ideal
			3.2 Canonical Form and Gröbner Bases of JC
		4 Identifying Neural Codes with Unique Marked Reduced Gröbner Bases
			4.1 Data Preparation
			4.2 Main Iterations
		5 Discussion and Future Work
		References
	The Number of Gröbner Bases in Finite Fields (Research)
		1 Introduction
		2 Background
			2.1 Algebraic Geometry Preliminaries
			2.2 Ideals Over Finite Fields
		3 Data Sets with m=2 Points
		4 Data Sets with m=3 Points
		5 Geometric Observations for Larger Sets
		6 Upper Bound for the Number of Gröbner Bases
		7 Discussion
		Appendix
		References
Part IV Commutative Algebra
	Depth of Powers of Squarefree Monomial Ideals (Research)
		1 Introduction
		2 Background
		3 Depth of Powers of Squarefree Monomial Ideals
		References
	A Note on the Uniqueness of Zero-Divisor Graphswith Loops (Research)
		1 Introduction
		2 Direct Products of Rings of Integers Modulo Various n
			2.1 Upper Triangular Matrix Rings Over Finite Fields
		References
	Some Combinatorial Cases of the Three Matrix Analog of Gerstenhaber's Theorem (Research)
		1 Introduction
			1.1 Towards Double Extensions of Cyclic Modules
			1.2 Other Combinatorial Classes
		2 A Reformulation of the Gerstenhaber Problem for Double Extensions of Cyclic Modules
		3 Addressing the Gerstenhaber Problem for Double Extensions of Cyclic Modules in a Combinatorial Case
			3.1 The Case I(xr, y, z)
			3.2 The Case I(xr, y, z), and Finishing the Proof of Theorem 1
		4 Gluing Plane Partitions: Addressing the Gerstenhaber Problem for Some Two-Generated Combinatorial Modules
			4.1 Young Diagrams and Skew-Diagrams
			4.2 Background on Gluing
			4.3 The Gerstenhaber Problem Where We Glue Corners
		5 Addressing the Gerstenhaber Problem in the Monomial Ideal Case
		References
	Structure of Semigroup Rings (Survey)
		1 Notations
			1.1 Semigroup Rings Presented as Quotients of Polynomial Rings
		2 Resolutions of Semigroup Rings
		3 Glued Semigroups
		References
Part V Analysis, Probability, and PDEs
	Using Monte Carlo Particle Methods to Estimate and Quantify Uncertainty in Periodic Parameters (Research)
		1 Introduction
		2 Parameter Estimation and Monte Carlo Particle Methods
			2.1 Augmented Ensemble Kalman Filter
		3 Estimating Periodic Parameters and the Role of Uncertainty
		4 Numerical Example: External Voltage in FitzHugh–Nagumo
		5 Discussion
		References
	A Note on Singularity Formation for a Nonlocal Transport Equation (Research)
		1 Introduction
		2 One Dimensional α-Patch Model and Main Results
		3 Proofs of Main Results
			3.1 Proof of Theorem 1 (Local Existence and Uniqueness)
			3.2 Proof of Theorem 2 (Singularity Formation)
				3.2.1 Preliminaries
				3.2.2 Proof of Theorem 2
		References
	Prescribing Initial Values for the Sticky Particle System (Survey)
		1 Introduction
		2 The Method of Characteristics
		3 Sticky Particle Trajectories
		4 Large Particle Limit
		5 Instability
		6 Lagrangian Variables
		7 Kinetic Theory
		References
Part VI Topology
	Towards Directed Collapsibility (Research)
		1 Introduction
		2 Concurrent Programs and Directed Path Spaces
		3 Past Links as Obstructions
		4 The Relationship Between Past Links and Path Spaces
		5 Directed Collapsibility
		6 Discussion
		References
	Contact Open Books and Symplectic Lefschetz Fibrations (Survey)
		1 Introduction
		2 Open Books and Lefschetz Fibrations
			2.1 Open Book Decompositions
			2.2 Open Books vs. Contact Manifolds
			2.3 Lefschetz Fibrations
				2.3.1 What do Lefschetz Fibrations Have to Do with Open Book Decompositions?
		3 Iterated Planar Open Books and Lefschetz Fibrations
			3.1 Planarity in High Dimensions
			3.2 Obstructions to Iterated Planarity
			3.3 The Weinstein Conjecture
		References
Part VII Applied Mathematics
	A Robust Preconditioner for High-Contrast Problems (Research)
		1 Introduction
		2 Problem Formulation and Main Results
			2.1 Derivation of a Singular Saddle Point Problem
			2.2 Preconditioned System and Its Implementation
			2.3 Main Result: Block-Diagonal Preconditioner
		3 Proofs of Statements in Sect.2.3
		4 Numerical Results
		5 Conclusions
		Appendix
		References
	On the Dimension Reduction in Prestrained Elasticity (Survey)
		1 Introduction
		2 Background on Thin Films and Dimension Reduction in Prestrained Elasticity
			2.1 Γ-Convergence
			2.2 Notation
		3 Relevant Literature and Current Developments
		4 Relative Bending Energy for Weakly Prestrained Shells
		5 General Scaling Result
		6 Conclusions
		References
	Machine Learning in Crowd Flow Exit Data (Survey)
		1 Introduction
		2 Methods for Estimating Exit Times
			2.1 Feature Engineering: Sliding Window Approach
			2.2 Auto-encoders for Dimensionality Reduction
		3 Estimation of Exit Times
			3.1 Experiments: PCA vs Auto-encoder with Neural Networks
		4 Summary and Future Research Directions
		References
	The Matter of Shape: A computational Approach to Making in Architectural Heritage (Survey)
		1 Introduction
		2 Materiality of Geometric Patterns
			2.1 Specifications of Carved Material Applications
			2.2 Specifications of Assembled Material Applications
		3 Concluding Remarks
		References
Part VIII Math Education
	Being Research-Based and Research-Minded in Helping K-12 Mathematics Education (Survey)
		1 Why Conduct Education Research?
		2 Where to Start? Important Theories That can Guide New Research Projects
			2.1 Professional Development
			2.2 Pedagogical Content Knowledge
			2.3 Teacher Quality
			2.4 Social Cognitive Career Theory
		3 A Brief Overview of Selected RUSMP Research Findings
		4 Closing Remarks
		References
	The Rice University School Mathematics Project: Supporting Excellence in K-16 Mathematics Since 1987 (Survey)
		1 Introduction
		2 Mission and Goals
		3 Programs for Educators
		4 Opportunities for Students and Support for Parents
		5 Support to Schools, School Districts, and the Community
		6 Support for Rice University and Other Universities
		7 Research and Evaluation Efforts
		8 Conclusion
		9 Selected RUSMP Publications
			9.1 The Early Years
			9.2 The Middle Years
			9.3 Recent Years