A Mathematical Primer for Computational Data Sciences C. Bajaj September 2, 2018 2 Contents Introduction 9 1 Graphs, Triangulations and Complexes 11 1.1 Graph Theory . 11 1.2 Combinatorial vs. Embedded Graphs . 11 1.2.1 Network Theory . 12 1.2.2 Trees and Spanning Trees . 12 1.3 Topological Complexes . 12 1.3.1 Pointset Topology . 12 1.3.2 CW-complexes . 13 1.3.3 Morse Functions and the Morse-Smale Complex . 13 1.3.4 Signed Distance Function and Critical Points of Discrete Distance Functions . 14 1.3.5 Contouring Tree Representation . 16 1.4 Complementary space topology and geometry . 16 1.4.1 Detection of Tunnels and Pockets . 17 1.5 Primal and Dual Complexes . 20 1.5.1 Primal Meshes . 20 1.5.2 Dual Complexes . 21 1.6 Voronoi and Delaunay Decompositions . 22 1.6.1 Euclidean vs. Power distance. 23 1.6.2 Weighted Alpha Shapes . 25 1.7 Biological Applications . 26 1.7.1 Union of Balls Topology . 26 1.7.2 Meshing of Molecular Interfaces . 27 Summary . 31 References and Further Reading . 31 Exercises . 32 2 Sets, Functions and Mappings 33 2.1 Scalar, Vector and Tensor Functions . 33 2.2 Inner Products and Norms . 33 2.2.1 Vector Space . 34 2.2.2 Topological Space . 34 2.2.3 Metric Space . 34 2.2.4 Topological Vector Space . 35 2.2.5 Normed Space . 35 2.2.6 Inner Product Space . 37 2.3 Piecewise-defined Functions . 38 2.4 Homogeneous and Barycentric coordinates . 38 2.4.1 Homogeneous coordinates . 38 2.4.2 Barycentric coordinates . 41 3 4 CONTENTS 2.5 Polynomials, Piecewise Polynomials, Splines . 43 2.5.1 Univariate case . 43 2.5.2 Bivariate case . 44 2.5.3 Multivariate case . 44 2.6 Parametric and Implicit Representation . 45 2.6.1 Curves . 45 2.6.2 Surface . 47 2.6.3 Examples . 47 2.7 Finite Elements and Error Estimation . 50 2.7.1 Tensor Product Over The Domain: Irregular Triangular Prism . 50 2.7.2 Generalized Barycentric Coordinate and Serendipity Elements . 53 2.7.3 Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes . 71 2.8 Biological Applications . 85 2.8.1 Tertiary Motif Detection . 85 2.8.2 Ion channel models . 85 2.8.3 Ribosome models . 86 2.8.4 Topological Agreement of Reduced Models . 86 2.8.5 Dynamic Deformation Visualization . 86 Summary . 87 References and Further Reading . 87 Exercises . 87 3 Differential Geometry, Operators 93 3.1 Shape Operators, First and Second Fundamental Forms . 93 3.1.1 Curvature: Gaussian, Mean . 93 3.1.2 The Shape of Space: convex, planar, hyperbolic . 93 3.1.3 Laplacian Eigenfunctions . 93 3.2 Finite Element Basis, Functional Spaces, Inner Products . 93 3.2.1 Hilbert Complexes . 93 3.3 Topology of Function Spaces . 93 3.4 Differential Operators and their Discretization formulas . 93 3.5 Conformal Mappings from Intrinsic Curvature . 94 3.6 Biological Applications . 95 3.6.1 Molecular Surface Analysis . 95 3.6.2 Solving PDEs in Biology . 95 Summary . 95 References and Further Reading . 95 Exercises . 95 4 Differential Forms and Homology of Discrete Functions 97 4.1 Exterior Calculus . 97 4.2 deRham Cohomology . 98 4.3 k-forms and k-cochains . 98 4.3.1 Discrete Differential Forms . ..