Posts

2023

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

Minimax Strategy
2022 Mar | Repeatedly choosing the action with the best worst-case scenario.

Backpropagation
2022 Feb | A convenient technique for computing gradients in neural networks.

Decision Trees
2022 Feb | We can algorithmically build classifiers that use a sequence of nested "if-then" decision rules.

Naive Bayes
2022 Jan | A simple classification algorithm grounded in Bayesian probability.

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.

Euler Estimation
2021 Mar | Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation.

K-Means Clustering
2021 Mar | Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive.

Basic Matrix Arithmetic
2021 Mar | We can use arrays to implement matrices and their associated mathematical operations.

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.

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.

Cartesian Product
2021 Jan | Implementing the Cartesian product provides good practice working with arrays.

Simulating Coin Flips
2021 Jan | Estimating probabilities by simulating a large number of random experiments.

2020

2019

Thales’ Theorem
2019 Dec | Every inscribed triangle whose hypotenuse is a diameter is a right triangle.

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.

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.

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.

Cutting Through the Hype of AI
2019 May | Media outlets often make the mistake of anthropomorphizing or attributing human-like characteristics to computer programs.

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.

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.

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.

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.

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.

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.

Radical Functions
2018 Jun | Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general.

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.

Zeros of Polynomials
2018 Apr | The zeros of a polynomial are the inputs that cause it to evaluate to zero.

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.

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.

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.

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.

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 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 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 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: 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.

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.

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 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.

2016

2015

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.

2014

2013

Sound Waves
2013 May | A brief overview of sound waves and how they interact with things.

Detecting Dark Matter
2013 May | A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP).

Evidence for the Existence of Dark Matter
2013 May | Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis's impact on baryon density.