Cover -- Title Page -- Copyright Page -- Contents -- List of Figures -- List of Tables -- Preface -- Acknowledgments -- Prologue -- Introductory Remarks -- Part I: Morphological Measurement in Macroevolutionary Distribution Analysis -- Chapter 1 Diatom Bauplan, As Modified 2D Valve Face Shapes of a 3D Capped Cylinder and Valve Shape Distribution -- 1.1 Introduction -- 1.1.1 Analytical Valve Shape Geometry -- 1.1.2 Valve Shape Constructs of Diatom Genera -- 1.2 Methods: A Test of Recurrent Diatom Valve Shapes -- 1.2.1 Legendre Polynomials, Hypergeometric Distribution, and Probabilities of Valve Shapes -- 1.2.2 Multivariate Hypergeometric Distribution of Diatom Valve Shapes as Recurrent Forms -- 1.3 Results -- 1.4 Discussion -- 1.4.1 Valve Shape Probability Distribution -- 1.4.2 Hypergeometric Functions and Other Shape Outline Methods -- 1.4.3 Application: Valve Shape Changes and Diversity during the Cenozoic -- 1.4.4 Diatom Valve Shape Distribution: Other Potential Studies -- 1.5 Summary and Future Research -- 1.6 Appendix -- 1.7 References -- Chapter 2 Comparative Surface Analysis and Tracking Changes in Diatom Valve Face Morphology -- 2.1 Introduction -- 2.1.1 Image Matching of Surface Features -- 2.1.2 Image Matching: Diatoms -- 2.2 Purpose of this Study -- 2.3 Background on Image and Surface Geometry -- 2.3.1 The Geometry of the Digital Image and the Jacobian -- 2.3.2 The Geometry of the Diatom 3D Surface Model and the Jacobian -- 2.3.3 The Image Gradient and Jacobian -- 2.4 Image Matching Kinematics via the Jacobian -- 2.4.1 Position and Motion: The Kinematics of Image Matching -- 2.4.2 Displacement and Implicit Functions -- 2.4.3 Displacement and Motion: Position and Orientation -- 2.4.4 Surface Feature Matching via the Jacobian -- 2.4.5 The Jacobian of Whole Surface Matching -- 2.5 Methods.

2.5.1 Fiducial Outcomes of Image Matching of Surface Features -- 2.6 Results -- 2.6.1 Surface Feature Image Matching and the Jacobian -- 2.6.2 Whole Valve Images, Matching of Crest Lines and the Jacobian -- 2.6.3 Image Matching of more than Two Images -- 2.7 Discussion -- 2.7.1 Utility of Jacobian-Based Methods and Image Matching -- 2.7.2 The Image Jacobian and Rotation in A Reference Frame: Potential Application to Diatom Images -- 2.7.3 Deformation and Registration of Image Surfaces: An Alternative Jacobian Calculation -- 2.8 Summary and Future Research -- 2.9 References -- Chapter 3 Diatom Valve Morphology, Surface Gradients and Natural Classification -- 3.1 Introduction -- 3.2 Purposes of this Study -- 3.2.1 The Genus Navicula -- 3.3 Methods -- 3.3.1 Naviculoid Diatom Surface Analysis -- 3.3.2 Gradients of Digital Image Surfaces -- 3.3.3 Histogram of Oriented Gradients and Surface Representation -- 3.3.4 Application to Diatom Valve Face Digital Images -- 3.3.5 Support Vector Regression and Classification -- 3.3.6 Using HOG as Combination Gradient Magnitude and Direction Input Data for SVR -- 3.3.7 Computational Efficiency and Cost -- 3.4 Diatom Valve Surface Morphological Analysis -- 3.4.1 SVR Model Fit of Naviculoid Taxa -- 3.4.2 Valve Surface Morphological Classification and Regression of Naviculoid Diatoms -- 3.5 Results -- 3.5.1 HOG Data Analysis -- 3.5.2 Goodness-of-Fit SVR Model Using 4D HOG Data from Naviculoid LMs -- 3.5.3 SVR of Naviculoid 2D HOG Data -- 3.5.4 Second Round of SVR Analysis of Remaining Naviculoid 2D HOG Data -- 3.5.5 Last Round of SVR Analysis of Remaining Naviculoid 2D HOG Data -- 3.5.6 Classification Results from SVR Analysis of Naviculoid Taxa -- 3.6 Discussion -- 3.6.1 Characteristics of SVM and SVR -- 3.6.2 Advantages in Using HOG Data and SVR -- 3.6.3 Potential Utility of HOG Data and SVR in Diatom Research.

3.7 Summary and Future Research -- 3.8 References -- Part II: Macroevolutionary Systems Analysis of Diatoms -- Chapter 4 Probabilistic Diatom Adaptive Radiation in the Southern Ocean -- 4.1 Introduction -- 4.1.1 Diatoms in the SO -- 4.1.2 Chaetocerotales and Bacillariales Speciation Rates -- 4.1.3 Chaetocerotales and Bacillariales: Fe, NO3 and SiO2 Availability in the SO -- 4.1.4 Modeling Diatom Adaptive Radiation -- 4.2 Purposes of this Study -- 4.3 Mathematical Modeling of Adaptive Radiation -- 4.3.1 Quantitative Phenotypic Trait Measurement and Adaptive Radiation -- 4.3.2 Adaptive Radiation: Implicit Stochastic Models -- 4.3.3 Adaptive Radiation Models: Time Evolution of a Stochastic System -- 4.3.4 Adaptive Radiation as an Optimal Control Problem -- 4.3.5 Exit Probabilities as Boundaries for Completion of Adaptive Radiation -- 4.3.6 Exit Times for the Adaptive Radiation Process -- 4.3.7 Adaptive Radiation: A Study of Southern Ocean Diatoms and Niche Filling -- 4.4 Methods -- 4.4.1 Niche Filling and Adaptive Radiation -- 4.5 Results -- 4.5.1 Ecological Niche Preference, Photosynthesis Efficiency, Nutrient Enrichment or Limitation, and Adaptive Radiation -- 4.5.2 Ecological Niche Preference and Photosynthesis Efficiency Rankings Representing Niche Filling as Adaptive Radiation -- 4.5.3 Niche Filling and the Lyapunov modified OU Adaptive Radiation Model -- 4.6 Discussion -- 4.6.1 More on Specifications for Adaptive Radiation Modeling -- 4.6.2 Diatom Adaptive Radiation Short-Term Trends as a Result of Niche Filling in the SO -- 4.6.3 Other Potential Mathematical Modeling Regimes of Adaptive Radiation -- 4.7 Summary and Future Directions -- 4.7.1 Caveats in Adaptive Radiation Studies to be Considered -- 4.8 References -- Chapter 5 Cenozoic Diatom Origination and Extinction and Influences on Diversity -- 5.1 Introduction.

5.1.1 Cenozoic Diatoms and Environmental Conditions -- 5.1.2 Diatom Diversity during the Cenozoic -- 5.1.3 Diversity as a Result of the Frequency of Origination and Extinction Events -- 5.2 Purposes of this Study -- 5.3 Methods and Background -- 5.3.1 Reconstructed Diatom Origination, Extinction and Diversity during the Cenozoic -- 5.3.2 Cumulative Functions and the Frequency of Cenozoic Origination and Extinction -- 5.3.3 Origination and Extinction: Heaviside Functions and Switching -- 5.3.4 Origination and Extinction as a Sequence of Steps and Accumulated Switches during the Cenozoic -- 5.3.5 Piecewise Continuous Switching via the Laplace Transform of the Heaviside Functions -- 5.3.6 Overlapping of Origination and Extinction: A Convolution Product -- 5.3.7 Non-Overlapping Origination and Extinction: A Poisson Process -- 5.3.8 Test of Switch Reversibility, Cenozoic Events and a Lyapunov Function -- 5.3.9 Origination and Extinction: Relation to Diversity -- 5.4 Results -- 5.4.1 Cumulative Frequency of Cenozoic Diatom Origination and Extinction Events -- 5.4.2 Switching from Diatom Origination to Extinction over the Cenozoic -- 5.4.3 Origination and Extinction Sequential Steps and Accumulated Switches during the Cenozoic -- 5.4.4 Overlapping via a Convolution Product of Origination and Extinction -- 5.4.5 Origination and Extinction as Poisson Processes -- 5.4.6 Test of Origination and Extinction Switches: Stochastic or Deterministic Chaos? -- 5.4.7 Diversity and Its Relation to Origination and Extinction for Cenozoic Diatoms -- 5.5 Discussion -- 5.5.1 Diversity and the Effects from Origination and Extinction of Cenozoic Diatoms -- 5.5.2 Cenozoic Events and Diatom Diversity, Origination and Extinction -- 5.5.3 Origination and Extinction Related to Diversity: Markov Chain, Martingale, Ergodic Processes, and Lyapunov Functions.

5.6 Summary and Future Research -- 5.7 References -- Chapter 6 Diatom Food Web Dynamics and Hydrodynamics Influences in the Arctic Ocean -- 6.1 Introduction -- 6.2 Purposes of this Study -- 6.3 Background on Arctic Ocean Diatoms -- 6.3.1 Diatoms and their Relation to Sea Ice -- 6.3.2 Sea Ice, Upwelling and Diatom Productivity -- 6.3.3 Diatom Lipid Content as a Proxy for Biomass -- 6.3.4 Diatom Biomass and the Hydrodynamics of Upwelling -- 6.4 Lattice Boltzmann Model -- 6.5 Lattice Boltzmann Model and Hydrodynamics -- 6.5.1 Upwelling and Buoyancy -- 6.5.2 Collisions and Streaming Densities of Diatom Genera during Upwelling -- 6.5.3 Buoyancy and Ice -- 6.5.4 Upwelling and the Splitting of the Cylindrical Rotation of Currents -- 6.6 Lattice Boltzmann Model: Diatom Bloom Density, Sea Ice and Upwelling -- 6.7 Lattice Boltzmann Model: Specifications for Simulation -- 6.7.1 Overview of 2D LBM with Respect to Diatom Genera Lattice Nodes -- 6.7.2 Buoyancy, Upwelling and Diatom Blooms in LBM via ñ and u -- 6.8 Methods -- 6.9 Results -- 6.10 Discussion -- 6.10.1 Arctic Diatom Food Web Dynamics: Other Potential Outcomes -- 6.10.2 Diatom Blooms: Influences over Time and Space -- 6.11 Summary and Future Research -- 6.12 References -- Part III: General and Special Functions in Diatom Macroevolutionary Spaces -- Chapter 7 Diatom Clade Biogeography: Climate Influences, Phenotypic Integration and Endemism -- 7.1 Introduction -- 7.1.1 Biogeography and Climate -- 7.1.2 Mapping Biogeographic Patterns -- 7.1.3 Biogeography as an Optimization Problem -- 7.1.4 Biogeographic Pattern and Spatial Rate of Change -- 7.1.5 Biogeography, Phenotypic Integration and Phenotypic Novelty -- 7.2 Purposes of this Study -- 7.3 Methods -- 7.3.1 Freshwater Diatom Dispersal Biogeography and the Traveling Salesman Problem.