Introduction to Algorithms and Machine Learning • Linear Algebra • Calculus • Algebra • Shorts

I used to write math books for fun as a way to consolidate and clarify my quantitative intuition. The goal was to provide deep intuition for the core concepts and connections, along with plenty of examples and exercises, while remaining as concise as possible.

A special thanks to Sanjana Kulkarni for her thoughtful suggestions and diligent proofreading of these books.

(expected 2023)   school program, pdf/print in progress

• Preamble: The Story of Math Academy's Eurisko Sequence
• 1. Hello World - Some Short Introductory Coding Exercises; Converting Between Binary, Decimal, and Hexadecimal; Recursive Sequences; Simulating Coin Flips; Roulette Wheel Selection; Cartesian Product.
• 2. Searching and Sorting - Brute Force Search with Linear-Encoding Cryptography; Solving Magic Squares via Backtracking; Estimating Roots via Bisection Search and Newton-Raphson Method; Single-Variable Gradient Descent; Multivariable Gradient Descent; Selection, Bubble, Insertion, and Counting Sort; Merge Sort and Quicksort.
• 3. Objects - Basic Matrix Arithmetic; Reduced Row Echelon Form and Applications to Matrix Arithmetic; K-Means Clustering; Tic-Tac-Toe and Connect Four; Euler Estimation; SIR Model for the Spread of Disease; Hodgkin-Huxley Model of Action Potentials in Neurons; Hash Tables; Simplex Method.
• 4. Regression and Classification - Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse; Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse; Power, Exponential, and Logistic Regression via Pseudoinverse; Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff; Regression via Gradient Descent; Multiple Regression and Interaction Terms; K-Nearest Neighbors; Naive Bayes.
• 5. Graphs - Breadth-First and Depth-First Traversals; Distance and Shortest Paths in Unweighted Graphs; Dijkstra's Algorithm for Distance and Shortest Paths in Weighted Graphs; Decision Trees; Introduction to Neural Network Regressors; Backpropagation.
• 6. Games - Canonical and Reduced Game Trees for Tic-Tac-Toe; Minimax Strategy; Reduced Search Depth and Heuristic Evaluation for Connect Four; Introduction to Blondie24 and Neuroevolution; Reimplementing Fogel's Tic-Tac-Toe Paper; Reimplementing Blondie24; Reimplementing Blondie24: Convolutional Version.

(2019)   pdf, print

• 1. Vectors - N-Dimensional Space; Dot Product and Cross Product; Lines and Planes; Span, Subspaces, and Reduction; Elimination as Vector Reduction.
• 2. Volume - N-Dimensional Volume Formula; Volume as the Determinant of a Square Linear System; Shearing, Cramer's Rule, and Volume by Reduction; Higher-Order Variation of Parameters.
• 3. Matrices - Linear Systems as Transformations of Vectors by Matrices; Matrix Multiplication; Rescaling, Shearing, and the Determinant; Inverse Matrices.
• 4. Eigenspace - Eigenvalues, Eigenvectors, and Diagonalization; Recursive Sequence Formulas via Diagonalization; Generalized Eigenvectors and Jordan Form; Matrix Exponential and Systems of Linear Differential Equations.

(2019)   pdf, print

• 1. Limits and Derivatives - Evaluating Limits; Limits by Logarithms, Squeeze Theorem, and Euler's Consant; Derivatives and the Difference Quotient; Power Rule; Chain Rule; Properties of Derivatives; Derivatives of Non-Polynomial Functions; Finding Local Extrema; Differentials and Approximation; L'Hôpital's Rule.
• 2. Integrals - Antiderivatives; Finding Area; Substitution; Integration by Parts; Improper Integrals.
• 3. Differential Equations - Separation of Variables; Slope Fields and Euler Approximation; Substitution; Characteristic Polynomial; Undetermined Coefficients; Integrating Factors; Variation of Parameters.
• 4. Series - Geometric Series; Tests for Convergence; Taylor Series; Manipulating Taylor Series; Solving Differential Equations with Taylor Series.

(2018)   pdf, print

• 1. Linear Equations and Systems - Solving Linear Equations; Slope-Intercept Form; Point-Slope Form; Standard Form; Linear Systems.
• 2. Quadratic Equations - Standard Form; Factoring; Quadratic Formula; Completing the Square; Vertex Form; Quadratic Systems.
• 3. Inequalities - Linear Inequalities in the Number Line; Linear Inequalities in the Plane; Quadratic Inequalities; Systems of Inequalities.
• 4. Polynomials - Standard Form and End Behavior; Zeros; Rational Roots and Synthetic Division; Sketching Graphs.
• 5. Rational Functions - Polynomial Long Division; Horizontal Asymptotes; Vertical Asymptotes; Graphing with Horizontal and Vertical Asymptotes; Graphing with Slant and Polynomial Asymptotes.
• 6. Non-Polynomial Functions - Radical Functions; Exponential and Logarithmic Functions; Absolute Value; Trigonometric Functions; Piecewise Functions.
• 7. Transformations of Functions - Shifts; Rescalings; Reflections; Inverse Functions; Compositions.

Below are some shorter manuscripts that I feel are interesting enough to share.

A Primer on Artificial Intelligence (2019)
What is AI; The First Wave: Reasoning as Search; The Second Wave: Expert Systems; The Third Wave: Computation Power and Neural Networks; Cutting Through the Hype.

Introduction to Python Programming (2019)
Getting Started in Colab; Strings, Ints, Floats, and Booleans; Lists, Dictionaries, and Arrays; If, While, and For; Functions.

Graphing Calculator Drawing Exercises (2019)

• 1. Lines - Horizontal and Vertical Lines; Slanted Lines; Absolute Value.
• 2. Open Curves - Parabolas; Sine Waves; Roots.
• 3. Closed Curves - Shading with Sine; Euclidean Ellipses, Non-Euclidean Ellipses.
• 4. Trigonometry - Rotation; Lissajous Curves; Composition Waves and Implicit Trig Patterns.

Intuiting Predictive Algorithms (2018)
Naive Bayes; MAP and MLE; Linear Regression; Support Vector Machines; Neural Networks; Decision Trees; Ensemble Models.

The Data Scientist's Guide to Topological Data Analysis (2017)

• Preamble
• 1. Mapper - Algorithm; Software; Use-Cases at Ayasdi; Use-Cases at Aunalytics.
• 2. Persistent Homology - Homotopy; Approximation; Homology; Persistence; Software.

Connecting Calculus to the Real World (2017)

• 1. Science and Medicine - Cardiac Output; Understanding Plaque Buildup; Modeling Tumor Growth.
• 2. Technology and Engineering - Rocket Propulsion; Rendering 3D Computer Graphics; Physics Engines in Video Games; Optimization via Gradient Descent.
• 3. Business and Economics - Maximizing Profit; Continuously Compounded Interest.
• 4. History and Philosophy - The Man who "Broke" Math; The Newton-Leibniz Controversy; A Failure of Intuition.
• 5. Art and Athletics - Derivatives in String Art; Calculating the Horsepower of an Offensive Lineman.

An Intuitive Primer on Calculus (2017)
Functions; Limits; Derivatives; Integrals; Sequences; Series.

The Physics Behind an Egg Drop: A Lively Story (2014)
Velocity; Momentum; Changes in Momentum; Force; Pressure; Troll Egg Drop.