What Mathematics Can Teach Us About Human Nature
2024 Apr | It highlights the aversion that people have to doing hard things. People will do unbelievable mental gymnastics to convince themselves that doing an easy, enjoyable thing that is unrelated to their supposed goal somehow moves the needle more than doing a hard, unpleasant thing that is directly related to said goal.
Posts
2024
Estimating a Visitation Interval: an Exercise in Bivariate Bayesian Statistics
2024 Mar | Loosely inspired by the German tank problem: several witnesses reported seeing a UFO during the given time intervals, and you want to quantify your certainty regarding when the UFO arrived and when it left.
Q&A: How to Grade Partial Credit
2024 Mar
Recreational Mathematics: Why Focus on Projects Over Puzzles
2024 Feb | There's only so much you can hone your math skills by working on a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way.
Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment
2024 Feb | The network becomes book-smart in a particular area but not street-smart in general. The training procedure is like a series of exams on material within a tiny subject area (your data subspace). The network refines its knowledge in the subject area to maximize its performance on those exams, but it doesn't refine its knowledge outside that subject area. And that leaves it gullible to adversarial examples using inputs outside the subject area.
Should Students be Asked to Regurgitate Known Proofs?
2024 Feb | Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.
Subtle Things to Watch Out For When Demonstrating Lp-Norm Regularization on a High-Degree Polynomial Regression Model
2024 Feb | Initial parameter range, data sampling range, severity of regularization.
The Only Way to Teach a More Sophisticated Technique
2024 Feb | ... is to present a problem where known simpler techniques fail.
How I Got Started with Calisthenics
2024 Jan | My training has been scattered and fuzzy until recently. Here's the whole story.
Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code
2024 Jan | I'd start off with some introductory course that covers the very basics of coding in some language that is used by many professional programmers but where the syntax reads almost like plain English and lower-level details like memory management are abstracted away. Then, I'd jump right into building board games and strategic game-playing agents (so a human can play against the computer), starting with simple games (e.g. tic-tac-toe) and working upwards from there (maybe connect 4 next, then checkers, and so on).
Tips for Learning Math Effectively
2024 Jan | Solving problems, building on top of what you've learned, reviewing what you've learned, and quality, quantity, and spacing of practice.
The Easiest Way to Remember Closed vs Open Interval Notation
2024 Jan | An oval () fits inside a rectangle [ ] with the same width and height.
Q&A: Why is Newton’s Method Useful?
2024 Jan
A Common Source of Student Mistakes
2024 Jan | Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results.
2023
Q&A: Undefined versus Infinity
2023 Dec
Critique of Paper: An astonishing regularity in student learning rate
2023 Nov | It rests on a critical assumption that the amount of learning that occurs during initial instruction is zero or otherwise negligible, which is not true.
My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?
2023 Nov | Q: Draw a 10 x 10 square grid. How many squares are there in total? Not just 1 x 1 squares, but also 2 x 2 squares, 3 x 3 squares, and so on. A: The total number of square shapes is the total sum of square numbers 1 + 4 + 9 + 16 + ... + 100.
Study Sessions Should be Short and Frequent as Opposed to Long and Sparse
2023 Nov | First, you want to form a habit. Second, you want to operate at peak productivity during your session. Third, you want to minimize the amount you forget between sessions.
Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?
2023 Oct | Answer: It's not very useful (not in practice, not in theory).
Can You Automate a Math Teacher?
2023 Oct | For many (but not all) students, the answer is yes. And for many of those students, automation can unlock life-changing educational outcomes.
The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning
2023 Oct | Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it's not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible.
When Can You Manipulate Differentials Like Fractions?
2023 Oct | In general, you can manipulate total derivatives like fractions, but you can't do the same with partial derivatives.
Q&A: Why are the Segment Addition Postulate and the Partition Postulate Two Different Things?
2023 Oct
How I Won a Heat Capacitor Competition Without a Heat Capacitor
2023 Oct | Won first place in a state-level competition by finding and exploiting a loophole in the points scoring logic.
How to Look Up the Meaning of an Unknown Math Symbol or Expression
2023 Sep | Drawing --> Latex commands --> ChatGPT summary --> Google more info
Q&A: How to Introduce Sigma Notation
2023 Sep
Q&A: Transitioning to One-Arm Chinup
2023 Sep
For Most Students, Competition Math is a Waste of Time
2023 Sep | Competition math problems generally don't require students to learn new fields of math. Rather, the difficulty comes from students needing to find clever tricks and insights to arrive at solutions using the mathematical tools that they have already learned. But if you look at the kinds of math that most quantitative professionals use on a daily basis, competition math tricks don't show up anywhere. But what does show up everywhere is university-level math subjects.
According to Feynman himself, his classes were a failure for 90% of his students.
2023 Sep | While some may view Feynman-style pedagogy as supporting inclusive learning for all students across varying levels of ability, Feynman himself acknowledged that his methods only worked for the top 10% of his students.
Q&A: Why is Zero Considered Even?
2023 Aug
Business Lessons from Science Fair
2023 Jun | The most important things I learned from competing in science fairs had nothing to do with physics or even academics. My main takeaways were actually related to business -- in particular, sales and marketing.
How to Remember Type I, II, and III Regions in Multivariable Calculus
2023 May | Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.
The Story of Math Academy’s Eurisko Sequence: the Most Advanced High School Math/CS Sequence in the USA
2023 May | During its operation from 2020 to 2023, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python).
Minimalist Strength Training, Phase 2: Gaining Mass
2023 Apr | Minor changes to increase workout intensity and caloric surplus.
My Experience with Teacher Credentialing and Professional Development
2023 Apr | It's centered around political ideology rather than the science of learning.
Why I Don’t Worship at the Altar of Neural Nets
2023 Jan | In order to justify using a more complex model, the increase in performance has to be worth the cost of integrating and maintaining the complexity.
Selecting a Good Problem to Work On
2023 Jan | Good problem = intersection between your own interests/talents, the realm of what's feasible, and the desires of the external world.
2022
Minimalist Strength Training, Phase 1: Getting Ripped
2022 Oct | Daily 20-30 minute bedroom workout with gymnastic rings hanging from pull-up bar -- just as much challenge as weights, but inexpensive and easily portable.
Quants vs Systems Coders
2022 Oct | Two subtypes of coders that I watched students grow into.
Tips for Developing Valuable Models
2022 Oct | Stuff you don't find in math textbooks.
The 5 Breeds of Quants
2022 Aug | ... are summarized in the following table.
From Procedures to Objects
2022 Aug | An aha moment with object-oriented programming.
Reimplementing Blondie24: Convolutional Version
2022 Mar | Using convolutional layers to create an even better checkers player.
Reimplementing Blondie24
2022 Mar | Extending Fogel's tic-tac-toe player to the game of checkers.
Reimplementing Fogel’s Tic-Tac-Toe Paper
2022 Mar | Reimplementing the paper that laid the groundwork for Blondie24.
Introduction to Blondie24 and Neuroevolution
2022 Mar | A method for training neural networks that works even when training feedback is sparse.
Reduced Search Depth and Heuristic Evaluation for Connect Four
2022 Mar | Combining game-specific human intelligence (heuristics) and generalizable artificial intelligence (minimax on a game tree)
Minimax Strategy
2022 Mar | Repeatedly choosing the action with the best worst-case scenario.
Canonical and Reduced Game Trees for Tic-Tac-Toe
2022 Mar | Building data structures that represent all the possible outcomes of a game.
Backpropagation
2022 Feb | A convenient technique for computing gradients in neural networks.
Introduction to Neural Network Regressors
2022 Feb | The deeper or more "hierarchical" a computational graph is, the more complex the model that it represents.
Decision Trees
2022 Feb | We can algorithmically build classifiers that use a sequence of nested "if-then" decision rules.
Dijkstra’s Algorithm for Distance and Shortest Paths in Weighted Graphs
2022 Feb | Computing spatial relationships between nodes when edges no longer represent unit distances.
Distance and Shortest Paths in Unweighted Graphs
2022 Feb | Using traversals to understand spatial relationships between nodes in graphs.
Breadth-First and Depth-First Traversals
2022 Feb | Graphs show up all the time in computer science, so it's important to know how to work with them.
Naive Bayes
2022 Jan | A simple classification algorithm grounded in Bayesian probability.
K-Nearest Neighbors
2022 Jan | One of the simplest classifiers.
Multiple Regression and Interaction Terms
2022 Jan | In many real-life situations, there is more than one input variable that controls the output variable.
Regression via Gradient Descent
2022 Jan | Gradient descent can help us avoid pitfalls that occur when fitting nonlinear models using the pseudoinverse.
Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff
2022 Jan | Just because model appears to match closely with points in the data set, does not necessarily mean it is a good model.
Power, Exponential, and Logistic Regression via Pseudoinverse
2022 Jan | Transforming nonlinear functions so that we can fit them using the pseudoinverse.
Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse
2022 Jan | Exploring the most general class of functions that can be fit using the pseudoinverse.
Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse
2022 Jan | Using matrix algebra to fit simple functions to data sets.
2021
Simplex Method
2021 Mar | A technique for maximizing linear expressions subject to linear constraints.
Hash Tables
2021 Mar | Under the hood, dictionaries are hash tables.
Hodgkin-Huxley Model of Action Potentials in Neurons
2021 Mar | Implementing a differential equations model that won the Nobel prize.
SIR Model For the Spread of Disease
2021 Mar | A simple differential equations model that we can plot using multivariable Euler estimation.
Euler Estimation
2021 Mar | Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation.
Tic-Tac-Toe and Connect Four
2021 Mar | One of the best ways to get practice with object-oriented programming is implementing games.
K-Means Clustering
2021 Mar | Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive.
Reduced Row Echelon Form and Applications to Matrix Arithmetic
2021 Mar | You can use the RREF algorithm to compute determinants much faster than with the recursive cofactor expansion method.
Basic Matrix Arithmetic
2021 Mar | We can use arrays to implement matrices and their associated mathematical operations.
The Ultimate High School Computer Science Sequence: 9 Months In
2021 Feb | In 9 months, these students went from initially not knowing how to write helper functions to building a machine learning library from scratch.
Merge Sort and Quicksort
2021 Feb | Merge sort and quicksort are generally faster than selection, bubble, and insertion sort. And unlike counting sort, they are not susceptible to blowup in the amount of memory required.
Selection, Bubble, Insertion, and Counting Sort
2021 Feb | Some of the simplest methods for sorting items in arrays.
Multivariable Gradient Descent
2021 Feb | Just like single-variable gradient descent, except that we replace the derivative with the gradient vector.
Single-Variable Gradient Descent
2021 Feb | We take an initial guess as to what the minimum is, and then repeatedly use the gradient to nudge that guess further and further "downhill" into an actual minimum.
Estimating Roots via Bisection Search and Newton-Raphson Method
2021 Feb | Bisection search involves repeatedly moving one bound halfway to the other. The Newton-Raphson method involves repeatedly moving our guess to the root of the tangent line.
Solving Magic Squares via Backtracking
2021 Feb | Backtracking can drastically cut down the number of possibilities that must be checked during brute force.
Brute Force Search with Linear-Encoding Cryptography
2021 Feb | Brute force search involves trying every single possibility.
Cartesian Product
2021 Jan | Implementing the Cartesian product provides good practice working with arrays.
Roulette Wheel Selection
2021 Jan | How to sample from a discrete probability distribution.
Simulating Coin Flips
2021 Jan | Estimating probabilities by simulating a large number of random experiments.
Recursive Sequences
2021 Jan | Sequences where each term is a function of the previous terms.
Converting Between Binary, Decimal, and Hexadecimal
2021 Jan | There are other number systems that use more or fewer than ten characters.
Some Short Introductory Coding Exercises
2021 Jan | It's assumed that you've had some basic exposure to programming.
2020
Tips for LaTeX Math Formatting
2020 Sep | How to avoid some of the most common pitfalls leading to ugly LaTeX.
2019
Path Dependency in Multivariable Limits
2019 Dec | The behavior of a multivariable function can be highly specific to the path taken.
Thales’ Theorem
2019 Dec | Every inscribed triangle whose hypotenuse is a diameter is a right triangle.
But WHERE do the Taylor Series and Lagrange Error Bound even come from?!
2019 Dec | An intuitive derivation.
Trick to Apply the Chain Rule FAST - Peeling the Onion
2019 Dec | A simple mnemonic trick for quickly differentiating complicated functions.
Intuition Behind Completing the Square
2019 Dec | Hidden inside of every quadratic, there is a perfect square.
Matrix Exponential and Systems of Linear Differential Equations
2019 Sep | The matrix exponential can be defined as a power series and used to solve systems of linear differential equations.
Generalized Eigenvectors and Jordan Form
2019 Sep | Jordan form provides a guaranteed backup plan for exponentiating matrices that are non-diagonalizable.
Recursive Sequence Formulas via Diagonalization
2019 Sep | Matrix diagonalization can be applied to construct closed-form expressions for recursive sequences.
Eigenvalues, Eigenvectors, and Diagonalization
2019 Sep | The eigenvectors of a matrix are those vectors that the matrix simply rescales, and the factor by which an eigenvector is rescaled is called its eigenvalue. These concepts can be used to quickly calculate large powers of matrices.
Inverse Matrices
2019 Aug | The inverse of a matrix is a second matrix which undoes the transformation of the first matrix.
Rescaling, Shearing, and the Determinant
2019 Aug | Every square matrix can be decomposed into a product of rescalings and shears.
Matrix Multiplication
2019 Aug | How to multiply a matrix by another matrix.
Linear Systems as Transformations of Vectors by Matrices
2019 Aug | Matrices are vectors whose components are themselves vectors.
Higher-Order Variation of Parameters
2019 Jul | Solving linear systems can sometimes be a necessary component of solving nonlinear systems.
Shearing, Cramer’s Rule, and Volume by Reduction
2019 Jul | Shearing can be used to express the solution of a linear system using ratios of volumes, and also to compute volumes themselves.
Volume as the Determinant of a Square Linear System
2019 Jul | Rich intuition about why the number of solutions to a square linear system is governed by the volume of the parallelepiped formed by the coefficient vectors.
N-Dimensional Volume Formula
2019 Jul | N-dimensional volume generalizes the idea of the space occupied by an object. We can think about N-dimensional volume as being enclosed by N-dimensional vectors.
Elimination as Vector Reduction
2019 Jun | If we interpret linear systems as sets of vectors, then elimination corresponds to vector reduction.
Span, Subspaces, and Reduction
2019 Jun | The span of a set of vectors consists of all vectors that can be made by adding multiples of vectors in the set. We can often reduce a set of vectors to a simpler set with the same span.
Lines and Planes
2019 Jun | A line starts at an initial point and proceeds straight in a constant direction. A plane is a flat sheet that makes a right angle with some particular vector.
Dot Product and Cross Product
2019 Jun | What does it mean to multiply a vector by another vector?
N-Dimensional Space
2019 Jun | N-dimensional space consists of points that have N components.
CheckMySteps: A Web App to Help Students Fix their Algebraic Mistakes
2019 May | A prototype web app to automatically assist students in self-correcting small errors and minor misconceptions.
Solving Tower of Hanoi with General Problem Solver
2019 May | A walkthrough of solving Tower of Hanoi using the approach of one of the earliest AI systems.
Cutting Through the Hype of AI
2019 May | Media outlets often make the mistake of anthropomorphizing or attributing human-like characteristics to computer programs.
The Third Wave of AI: Computation Power and Neural Networks
2019 May | As computation power increased, neural networks began to take center stage in AI.
The Second Wave of AI: Expert Systems
2019 May | Expert systems stored "if-then" rules derived from the knowledge of experts.
The First Wave of AI: Reasoning as Search
2019 May | Framing reasoning as searching through a maze of actions for a sequence that achieves the desired end goal.
What is AI?
2019 May | Turing test, games, hype, narrow vs general AI.
Introductory Python: Functions
2019 May | Rather than duplicating such code each time we want to use it, it is more efficient to store the code in a function.
Introductory Python: If, While, and For
2019 May | We often wish to tell the computer instructions involving the words "if," "while," and "for."
Introductory Python: Lists, Dictionaries, and Arrays
2019 May | We can store many related pieces of data within a single variable called a data structure.
Introductory Python: Strings, Ints, Floats, and Booleans
2019 May | We can store and manipulate data in the form of variables.
Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns
2019 May | Equations involving compositions of trigonometric functions can create wild patterns in the plane.
Graphing Calculator Drawing: Lissajous Curves
2019 May | Lissajous curves use sine functions to create interesting patterns in the plane.
Graphing Calculator Drawing: Rotation
2019 May | Absolute value graphs can be rotated to draw stars.
Graphing Calculator Drawing: Non-Euclidean Ellipses
2019 May | Non-euclidean ellipses can be used to draw starry-eye sunglasses.
Graphing Calculator Drawing: Euclidean Ellipses
2019 May | Euclidean ellipses can be combined with sine wave shading to form three-dimensional shells.
Graphing Calculator Drawing: Shading with Sine
2019 May | High-frequency sine waves can be used to draw shaded regions.
Graphing Calculator Drawing: Roots
2019 May | Roots can be used to draw deer.
Graphing Calculator Drawing: Sine Waves
2019 May | Sine waves can be used to draw scales on a fish.
Graphing Calculator Drawing: Parabolas
2019 May | Parabolas can be used to draw a fish.
Graphing Calculator Drawing: Absolute Value
2019 May | Absolute value can be used to draw a person.
Graphing Calculator Drawing: Slanted Lines
2019 May | Slanted lines can be used to draw a spider web.
Graphing Calculator Drawing: Horizontal and Vertical Lines
2019 May | Horizontal and vertical lines can be used to draw a castle.
Solving Differential Equations with Taylor Series
2019 Apr | Many differential equations don't have solutions that can be expressed in terms of finite combinations of familiar functions. However, we can often solve for the Taylor series of the solution.
Manipulating Taylor Series
2019 Apr | To find the Taylor series of complicated functions, it's often easiest to manipulate the Taylor series of simpler functions.
Taylor Series
2019 Apr | Many non-polynomial functions can be represented by infinite polynomials.
Tests for Convergence
2019 Apr | Various tricks for determining whether a series converges or diverges.
Geometric Series
2019 Apr | A geometric series is a sum where each term is some constant times the previous term.
Variation of Parameters
2019 Mar | When we know the solutions of a linear differential equation with constant coefficients and right hand side equal to zero, we can use variation of parameters to find a solution when the right hand side is not equal to zero.
Integrating Factors
2019 Mar | Integrating factors can be used to solve first-order differential equations with non-constant coefficients.
Undetermined Coefficients
2019 Mar | Undetermined coefficients can help us find a solution to a linear differential equation with constant coefficients when the right hand side is not equal to zero.
Characteristic Polynomial of a Differential Equation
2019 Mar | Given a linear differential equation with constant coefficients and a right hand side of zero, the roots of the characteristic polynomial correspond to solutions of the equation.
Solving Differential Equations by Substitution
2019 Mar | Non-separable differential equations can be sometimes converted into separable differential equations by way of substitution.
Slope Fields and Euler Approximation
2019 Mar | When faced with a differential equation that we don't know how to solve, we can sometimes still approximate the solution.
Separation of Variables
2019 Mar | The simplest differential equations can be solved by separation of variables, in which we move the derivative to one side of the equation and take the antiderivative.
Improper Integrals
2019 Feb | Improper integrals have bounds or function values that extend to positive or negative infinity.
Integration by Parts
2019 Feb | We can apply integration by parts whenever an integral would be made simpler by differentiating some expression within the integral, at the cost of anti-differentiating another expression within the integral.
Integration by Substitution
2019 Feb | Substitution involves condensing an expression of into a single new variable, and then expressing the integral in terms of that new variable.
Finding Area Using Integrals
2019 Feb | To evaluate a definite integral, we find the antiderivative, evaluate it at the indicated bounds, and then take the difference.
Antiderivatives
2019 Feb | The antiderivative of a function is a second function whose derivative is the first function.
L’Hôpital’s Rule
2019 Jan | When a limit takes the indeterminate form of zero divided by zero or infinity divided by infinity, we can differentiate the numerator and denominator separately without changing the actual value of the limit.
Differentials and Approximation
2019 Jan | We can interpret the derivative as an approximation for how a function's output changes, when the function input is changed by a small amount.
Finding Extrema
2019 Jan | Derivatives can be used to find a function's local extreme values, its peaks and valleys.
Derivatives of Non-Polynomial Functions
2019 Jan | There are convenient rules the derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions.
Properties of Derivatives
2019 Jan | Given a sum, we can differentiate each term individually. But why are we able to do this? Does multiplication work the same way? What about division?
Chain Rule
2019 Jan | When taking derivatives of compositions of functions, we can ignore the inside of a function as long as we multiply by the derivative of the inside afterwards.
Power Rule for Derivatives
2019 Jan | There are some patterns that allow us to compute derivatives without having to compute the limit of the difference quotient.
Derivatives and the Difference Quotient
2019 Jan | The derivative of a function is the function's slope at a particular point, and can be computed as the limit of the difference quotient.
Limits by Logarithms, Squeeze Theorem, and Euler’s Constant
2019 Jan | Various tricks for evaluating tricky limits.
Evaluating Limits
2019 Jan | The limit of a function, as the input approaches some value, is the output we would expect if we saw only the surrounding portion of the graph.
2018
Compositions of Functions
2018 Jul | Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function.
Inverse Functions
2018 Jul | Inverting a function entails reversing the outputs and inputs of the function.
Reflections of Functions
2018 Jul | When a function is reflected, it flips across one of the axes to become its mirror image.
Rescalings of Functions
2018 Jul | When a function is rescaled, it is stretched or compressed along one of the axes, like a slinky.
Shifts of Functions
2018 Jul | When a function is shifted, all of its points move vertically and/or horizontally by the same amount.
Piecewise Functions
2018 Jun | A piecewise function is pieced together from multiple different functions.
Trigonometric Functions
2018 Jun | Trigonometric functions represent the relationship between sides and angles in right triangles.
Absolute Value
2018 Jun | Absolute value represents the magnitude of a number, i.e. its distance from zero.
Exponential and Logarithmic Functions
2018 Jun | Exponential functions have variables as exponents. Logarithms cancel out exponentiation.
Radical Functions
2018 Jun | Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general.
Graphing Rational Functions with Slant and Polynomial Asymptotes
2018 May | A slant asymptote is a slanted line that arises from a linear term in the proper form of a rational function.
Graphing Rational Functions with Horizontal and Vertical Asymptotes
2018 May | If we choose one input on each side of an asymptote, we can tell which section of the plane the function will occupy.
Vertical Asymptotes of Rational Functions
2018 May | Vertical asymptotes are vertical lines that a function approaches but never quite reaches.
Horizontal Asymptotes of Rational Functions
2018 May | Rational functions can have a form of end behavior in which they become flat, approaching (but never quite reaching) a horizontal line known as a horizontal asymptote.
Polynomial Long Division
2018 May | Polynomial long division works the same way as the long division algorithm that's familiar from simple arithmetic.
Sketching Graphs of Polynomials
2018 Apr | We can sketch the graph of a polynomial using its end behavior and zeros.
Rational Roots and Synthetic Division
2018 Apr | The rational roots theorem can help us find zeros of polynomials without blindly guessing.
Zeros of Polynomials
2018 Apr | The zeros of a polynomial are the inputs that cause it to evaluate to zero.
Standard Form and End Behavior of Polynomials
2018 Apr | The end behavior of a polynomial refers to the type of output that is produced when we input extremely large positive or negative values.
Systems of Inequalities
2018 Mar | To solve a system of inequalities, we need to solve each individual inequality and find where all their solutions overlap.
Quadratic Inequalities
2018 Mar | Quadratic inequalities are best visualized in the plane.
Linear Inequalities in the Plane
2018 Mar | When a linear equation has two variables, the solution covers a section of the coordinate plane.
Linear Inequalities in the Number Line
2018 Mar | An inequality is similar to an equation, but instead of saying two quantities are equal, it says that one quantity is greater than or less than another.
Quadratic Systems
2018 Feb | Systems of quadratic equations can be solved via substitution.
Vertex Form
2018 Feb | To easily graph a quadratic equation, we can convert it to vertex form.
Completing the Square
2018 Feb | Completing the square helps us gain a better intuition for quadratic equations and understand where the quadratic formula comes from.
Quadratic Formula
2018 Feb | To solve hard-to-factor quadratic equations, it's easiest to use the quadratic formula.
Factoring Quadratic Equations
2018 Feb | Factoring is a method for solving quadratic equations.
Standard Form of a Quadratic Equation
2018 Feb | Quadratic equations are similar to linear equations, except that they contain squares of a single variable.
Linear Systems
2018 Feb | A linear system consists of multiple linear equations, and the solution of a linear system consists of the pairs that satisfy all of the equations.
Standard Form of a Line
2018 Feb | Standard form makes it easy to see the intercepts of a line.
Point-Slope Form
2018 Feb | An easy way to write the equation of a line if we know the slope and a point on a line.
Slope-Intercept Form
2018 Feb | Introducing linear equations in two variables.
Solving Linear Equations
2018 Feb | Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division.
Intuiting Ensemble Methods
2018 Jan | The type of ensemble model that wins most data science competitions is the stacked model, which consists of an ensemble of entirely different species of models together with some combiner algorithm.
Intuiting Decision Trees
2018 Jan | Decision trees are able to model nonlinear data while remaining interpretable.
Intuiting Neural Networks
2018 Jan | NNs are similar to SVMs in that they project the data to a higher-dimensional space and fit a hyperplane to the data in the projected space. However, whereas SVMs use a predetermined kernel to project the data, NNs automatically construct their own projection.
Intuiting Support Vector Machines
2018 Jan | A Support Vector Machine (SVM) computes the "best" separation between classes as the maximum-margin hyperplane.
Intuiting Linear Regression
2018 Jan | In linear regression, we model the target as a random variable whose expected value depends on a linear combination of the predictors (including a bias term).
Intuiting Maximum a Posteriori and Maximum Likelihood Estimation
2018 Jan | To visualize the relationship between the MAP and MLE estimations, one can imagine starting at the MLE estimation, and then obtaining the MAP estimation by drifting a bit towards higher density in the prior distribution.
Intuiting Naive Bayes
2018 Jan | Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature.
2017
Applications of Calculus: Calculating the Horsepower of an Offensive Lineman
2017 Nov | It comes out to roughly a fortieth of that of a truck.
Applications of Calculus: Derivatives in String Art
2017 Nov | String art works because the strings are tangent lines to a curve.
Applications of Calculus: A Failure of Intuition
2017 Nov | Calculus can show us how our intuition can fail us, a common theme in philosophy.
History of Calculus: The Newton-Leibniz Controversy
2017 Nov | Nobody came out of the dispute well.
History of Calculus: The Man who “Broke” Math
2017 Nov | When Joseph Fourier first introduced Fourier series, they gave mathematicians nightmares.
Applications of Calculus: Continuously Compounded Interest
2017 Nov | Deriving the "Pert" formula.
Applications of Calculus: Maximizing Profit
2017 Nov | If we know the revenue and costs associated with producing any number of units, then we can use calculus to figure out the number of units to produce for maximum profit.
Applications of Calculus: Optimization via Gradient Descent
2017 Nov | Calculus can be used to find the parameters that minimize a function.
Applications of Calculus: Physics Engines in Video Games
2017 Nov | Physics engines use calculus to periodically updates the locations of objects.
Applications of Calculus: Rendering 3D Computer Graphics
2017 Nov | Introducing Kajiya's rendering equation.
Applications of Calculus: Rocket Propulsion
2017 Nov | Deriving the ideal rocket equation.
Applications of Calculus: Modeling Tumor Growth
2017 Nov | Deriving the Gompertz function.
Applications of Calculus: Understanding Plaque Buildup
2017 Nov | Understanding why even slight narrowing of arteries can pose such a big problem to blood flow.
Applications of Calculus: Cardiac Output
2017 Nov | Measuring volume of blood the heart pumps out into the aorta per unit time.
Intuiting Series
2017 Oct | A series is the sum of a sequence.
Intuiting Sequences
2017 Oct | A sequence is a list of numbers that has some pattern.
Intuiting Integrals
2017 Oct | Integrals give the area under a portion of a function.
Intuiting Derivatives
2017 Oct | The derivative tells the steepness of a function at a given point, kind of like a carpenter's level.
Intuiting Limits
2017 Oct | The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line.
Intuiting Functions
2017 Oct | A function is a scribble that crosses each vertical line only once.
The Data Scientist’s Guide to Topological Data Analysis: Preamble
2017 Sep | Bridging the communication gap between academia and industry in the field of TDA.
Persistent Homology Software: Demonstration of TDA
2017 Sep | Demonstrating an open-source implementation of persistent homology techniques in the TDA package for R.
Intuiting Persistent Homology
2017 Sep | Persistent homology provides a way to quantify the topological features that persist over our a data set's full range of scale.
Mapper Use-Cases at Aunalytics
2017 Sep | At Aunalytics, Mapper outperformed hierarchical clustering in providing granular insights.
Mapper Use-Cases at Ayasdi
2017 Sep | Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data.
Mapper Software: Demonstration of TDAmapper
2017 Sep | Demonstrating an open-source implementation of Mapper in the TDAmapper package for R.
Intuiting the Mapper Algorithm
2017 Sep | Representing a data space's topology by converting it into a network.
2016
A Game-Theoretic Analysis of Social Distancing During Epidemics
2016 May | In a simplified problem framing, we investigate the (game-theoretical) usefulness of limiting the number of social connections per person.
Making Indirect Interactions Explicit in Networks
2016 Mar | Category theory provides a language for explicitly describing indirect relationships in graphs.
Book Summary: Memory Evolutive Systems
2016 Mar | Framing complex systems in the language of category theory.
2015
Introduction to Computers
2015 Nov | The main ideas behind computers can be understood by anyone.
The Brain in One Sentence
2015 Nov | The brain is a neuronal network integrating specialized subsystems that use local competition and thresholding to sparsify input, spike-timing dependent plasticity to learn inference, and layering to implement hierarchical predictive learning.
Shaping STDP Neural Networks with Periodic Stimulation: a Theoretical Analysis for the Case of Tree Networks
2015 Aug | We solve a special case of how to periodically stimulate a biological neural network to obtain a desired connectivity (in theory).
On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne
2015 May | Montaigne's education, strictly dictated by his parents and university studies, resulted in an isolative work with scholarly impact but limited public reach. Conversely, Benjamin Franklin's goal-oriented self-teaching led to influential creations and roles benefiting his community and nation.
A Brief Overview of Spike-Timing Dependent Plasticity (STDP) Learning During Neural Simulation
2015 Feb | Implementation notes for STDP learning in a network of Hodgkin-Huxley simulated neurons.
A Visual, Inductive Proof of Sharkovsky’s Theorem
2015 Jan | Many existing proofs are not accessible to young mathematicians or those without experience in the realm of dynamic systems.
2014
Building an Iron Man Suit: A Physics Workbook
2014 Jul | A workbook I created to explain the math and physics behind an Iron Man suit to a student who was interested in the comics / movies.
The Physics Behind an Egg Drop: A Lively Story
2014 Jul | A workbook I created to explain the math and physics behind an egg drop experiment to a student who was interested in Lord of the Rings and Star Wars.
2013
A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$
2013 Aug | And a proof via double induction.
2012
Sound Waves
2012 Dec | A brief overview of sound waves and how they interact with things.
Detecting Dark Matter
2012 Dec | A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP).
Evidence for the Existence of Dark Matter
2012 Dec | Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis's impact on baryon density.