# Sitemap

XML version

<font size="3em"><p class="archive__item-excerpt" itemprop="description"><p> <strong><a href="https://justinmath.com/2018SpringWaldenMathCircle/" rel="permalink"> Read more...</a></strong></p></p></font>
<!---->


</article> </div>

–>

## How to Remember Type I, II, and III Regions in Multivariable Calculus

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. Read more...

## Minimalist Strength Training, Phase 2: Gaining Mass

Minor changes to increase workout intensity and caloric surplus. Read more...

## My Experience with Teacher Credentialing and Professional Development

It’s centered around political ideology rather than the science of learning. Read more...

## The Story of Math Academy’s Eurisko Sequence

During its operation from 2020 to 2023, Eurisko was the most advanced high school math/CS sequence in the USA. Read more...

## Why I Don’t Worship at the Altar of Neural Nets

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. Read more...

## Selecting a Good Problem to Work On

Good problem = intersection between your own interests/talents, the realm of what’s feasible, and the desires of the external world. Read more...

## From Procedures to Objects

An aha moment with object-oriented programming. Read more...

## Minimalist Strength Training, Phase 1: Getting Ripped

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. Read more...

## Quants vs Systems Coders

Two subtypes of coders that I watched students grow into. Read more...

## Tips for Developing Valuable Models

Stuff you don’t find in math textbooks. Read more...

## The 5 Breeds of Quants

… are summarized in the following table. Read more...

## Reimplementing Blondie24: Convolutional Version

Using convolutional layers to create an even better checkers player. Read more...

## Reimplementing Blondie24

Extending Fogel’s tic-tac-toe player to the game of checkers. Read more...

## Reimplementing Fogel’s Tic-Tac-Toe Paper

Reimplementing the paper that laid the groundwork for Blondie24. Read more...

## Introduction to Blondie24 and Neuroevolution

A method for training neural networks that works even when training feedback is sparse. Read more...

## Reduced Search Depth and Heuristic Evaluation for Connect Four

Combining game-specific human intelligence (heuristics) and generalizable artificial intelligence (minimax on a game tree) Read more...

## Minimax Strategy

Repeatedly choosing the action with the best worst-case scenario. Read more...

## Canonical and Reduced Game Trees for Tic-Tac-Toe

Building data structures that represent all the possible outcomes of a game. Read more...

## Introduction to Neural Network Regressors

The deeper or more “hierarchical” a computational graph is, the more complex the model that it represents. Read more...

## Decision Trees

We can algorithmically build classifiers that use a sequence of nested “if-then” decision rules. Read more...

## Dijkstra’s Algorithm for Distance and Shortest Paths in Weighted Graphs

Computing spatial relationships between nodes when edges no longer represent unit distances. Read more...

## Distance and Shortest Paths in Unweighted Graphs

Using traversals to understand spatial relationships between nodes in graphs. Read more...

Graphs show up all the time in computer science, so it’s important to know how to work with them. Read more...

## Naive Bayes

A simple classification algorithm grounded in Bayesian probability. Read more...

## K-Nearest Neighbors

One of the simplest classifiers. Read more...

## Multiple Regression and Interaction Terms

In many real-life situations, there is more than one input variable that controls the output variable. Read more...

Gradient descent can help us avoid pitfalls that occur when fitting nonlinear models using the pseudoinverse. Read more...

## Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff

Just because model appears to match closely with points in the data set, does not necessarily mean it is a good model. Read more...

## Power, Exponential, and Logistic Regression via Pseudoinverse

Transforming nonlinear functions so that we can fit them using the pseudoinverse. Read more...

## Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse

Exploring the most general class of functions that can be fit using the pseudoinverse. Read more...

## Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse

Using matrix algebra to fit simple functions to data sets. Read more...

## Simplex Method

A technique for maximizing linear expressions subject to linear constraints. Read more...

## Hash Tables

Under the hood, dictionaries are hash tables. Read more...

## Hodgkin-Huxley Model of Action Potentials in Neurons

Implementing a differential equations model that won the Nobel prize. Read more...

## SIR Model For the Spread of Disease

A simple differential equations model that we can plot using multivariable Euler estimation. Read more...

## Euler Estimation

Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation. Read more...

## Tic-Tac-Toe and Connect Four

One of the best ways to get practice with object-oriented programming is implementing games. Read more...

## K-Means Clustering

Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive. Read more...

## Reduced Row Echelon Form and Applications to Matrix Arithmetic

You can use the RREF algorithm to compute determinants much faster than with the recursive cofactor expansion method. Read more...

## Basic Matrix Arithmetic

We can use arrays to implement matrices and their associated mathematical operations. Read more...

## The Ultimate High School Computer Science Sequence: 9 Months In

In 9 months, these students went from initially not knowing how to write helper functions to building a machine learning library from scratch. Read more...

## Merge Sort and Quicksort

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. Read more...

## Selection, Bubble, Insertion, and Counting Sort

Some of the simplest methods for sorting items in arrays. Read more...

Just like single-variable gradient descent, except that we replace the derivative with the gradient vector. Read more...

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. Read more...

## Estimating Roots via Bisection Search and Newton-Raphson Method

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. Read more...

## Solving Magic Squares via Backtracking

Backtracking can drastically cut down the number of possibilities that must be checked during brute force. Read more...

## Brute Force Search with Linear-Encoding Cryptography

Brute force search involves trying every single possibility. Read more...

## Cartesian Product

Implementing the Cartesian product provides good practice working with arrays. Read more...

## Roulette Wheel Selection

How to sample from a discrete probability distribution. Read more...

## Simulating Coin Flips

Estimating probabilities by simulating a large number of random experiments. Read more...

## Recursive Sequences

Sequences where each term is a function of the previous terms. Read more...

## Converting Between Binary, Decimal, and Hexadecimal

There are other number systems that use more or fewer than ten characters. Read more...

## Some Short Introductory Coding Exercises

It’s assumed that you’ve had some basic exposure to programming. Read more...

## Tips for LaTeX Math Formatting

How to avoid some of the most common pitfalls leading to ugly LaTeX. Read more...

## Path Dependency in Multivariable Limits

The behavior of a multivariable function can be highly specific to the path taken. Read more...

## Thales’ Theorem

Every inscribed triangle whose hypotenuse is a diameter is a right triangle. Read more...

## Trick to Apply the Chain Rule FAST - Peeling the Onion

A simple mnemonic trick for quickly differentiating complicated functions. Read more...

## Intuition Behind Completing the Square

Hidden inside of every quadratic, there is a perfect square. Read more...

## Matrix Exponential and Systems of Linear Differential Equations

The matrix exponential can be defined as a power series and used to solve systems of linear differential equations. Read more...

## Generalized Eigenvectors and Jordan Form

Jordan form provides a guaranteed backup plan for exponentiating matrices that are non-diagonalizable. Read more...

## Recursive Sequence Formulas via Diagonalization

Matrix diagonalization can be applied to construct closed-form expressions for recursive sequences. Read more...

## Eigenvalues, Eigenvectors, and Diagonalization

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. Read more...

## Inverse Matrices

The inverse of a matrix is a second matrix which undoes the transformation of the first matrix. Read more...

## Rescaling, Shearing, and the Determinant

Every square matrix can be decomposed into a product of rescalings and shears. Read more...

## Matrix Multiplication

How to multiply a matrix by another matrix. Read more...

## Linear Systems as Transformations of Vectors by Matrices

Matrices are vectors whose components are themselves vectors. Read more...

## Higher-Order Variation of Parameters

Solving linear systems can sometimes be a necessary component of solving nonlinear systems. Read more...

## Shearing, Cramer’s Rule, and Volume by Reduction

Shearing can be used to express the solution of a linear system using ratios of volumes, and also to compute volumes themselves. Read more...

## Volume as the Determinant of a Square Linear System

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. Read more...

## N-Dimensional Volume Formula

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. Read more...

## Elimination as Vector Reduction

If we interpret linear systems as sets of vectors, then elimination corresponds to vector reduction. Read more...

## Span, Subspaces, and Reduction

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. Read more...

## Lines and Planes

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. Read more...

## Dot Product and Cross Product

What does it mean to multiply a vector by another vector? Read more...

## N-Dimensional Space

N-dimensional space consists of points that have N components. Read more...

## CheckMySteps: A Web App to Help Students Fix their Algebraic Mistakes

A prototype web app to automatically assist students in self-correcting small errors and minor misconceptions. Read more...

## Solving Tower of Hanoi with General Problem Solver

A walkthrough of solving Tower of Hanoi using the approach of one of the earliest AI systems. Read more...

## Cutting Through the Hype of AI

Media outlets often make the mistake of anthropomorphizing or attributing human-like characteristics to computer programs. Read more...

## The Third Wave of AI: Computation Power and Neural Networks

As computation power increased, neural networks began to take center stage in AI. Read more...

## The Second Wave of AI: Expert Systems

Expert systems stored “if-then” rules derived from the knowledge of experts. Read more...

## The First Wave of AI: Reasoning as Search

Framing reasoning as searching through a maze of actions for a sequence that achieves the desired end goal. Read more...

## What is AI?

Turing test, games, hype, narrow vs general AI. Read more...

## Introductory Python: Functions

Rather than duplicating such code each time we want to use it, it is more efficient to store the code in a function. Read more...

## Introductory Python: If, While, and For

We often wish to tell the computer instructions involving the words “if,” “while,” and “for.” Read more...

## Introductory Python: Lists, Dictionaries, and Arrays

We can store many related pieces of data within a single variable called a data structure. Read more...

## Introductory Python: Strings, Ints, Floats, and Booleans

We can store and manipulate data in the form of variables. Read more...

## Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns

Equations involving compositions of trigonometric functions can create wild patterns in the plane. Read more...

## Graphing Calculator Drawing: Lissajous Curves

Lissajous curves use sine functions to create interesting patterns in the plane. Read more...

## Graphing Calculator Drawing: Rotation

Absolute value graphs can be rotated to draw stars. Read more...

## Graphing Calculator Drawing: Non-Euclidean Ellipses

Non-euclidean ellipses can be used to draw starry-eye sunglasses. Read more...

## Graphing Calculator Drawing: Euclidean Ellipses

Euclidean ellipses can be combined with sine wave shading to form three-dimensional shells. Read more...

## Graphing Calculator Drawing: Shading with Sine

High-frequency sine waves can be used to draw shaded regions. Read more...

## Graphing Calculator Drawing: Roots

Roots can be used to draw deer. Read more...

## Graphing Calculator Drawing: Sine Waves

Sine waves can be used to draw scales on a fish. Read more...

## Graphing Calculator Drawing: Parabolas

Parabolas can be used to draw a fish. Read more...

## Graphing Calculator Drawing: Absolute Value

Absolute value can be used to draw a person. Read more...

## Graphing Calculator Drawing: Slanted Lines

Slanted lines can be used to draw a spider web. Read more...

## Graphing Calculator Drawing: Horizontal and Vertical Lines

Horizontal and vertical lines can be used to draw a castle. Read more...

## Solving Differential Equations with Taylor Series

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. Read more...

## Manipulating Taylor Series

To find the Taylor series of complicated functions, it’s often easiest to manipulate the Taylor series of simpler functions. Read more...

## Taylor Series

Many non-polynomial functions can be represented by infinite polynomials. Read more...

## Tests for Convergence

Various tricks for determining whether a series converges or diverges. Read more...

## Geometric Series

A geometric series is a sum where each term is some constant times the previous term. Read more...

## Variation of Parameters

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. Read more...

## Integrating Factors

Integrating factors can be used to solve first-order differential equations with non-constant coefficients. Read more...

## Undetermined Coefficients

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. Read more...

## Characteristic Polynomial of a Differential Equation

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. Read more...

## Solving Differential Equations by Substitution

Non-separable differential equations can be sometimes converted into separable differential equations by way of substitution. Read more...

## Slope Fields and Euler Approximation

When faced with a differential equation that we don’t know how to solve, we can sometimes still approximate the solution. Read more...

## Separation of Variables

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. Read more...

## Improper Integrals

Improper integrals have bounds or function values that extend to positive or negative infinity. Read more...

## Integration by Parts

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. Read more...

## Integration by Substitution

Substitution involves condensing an expression of into a single new variable, and then expressing the integral in terms of that new variable. Read more...

## Finding Area Using Integrals

To evaluate a definite integral, we find the antiderivative, evaluate it at the indicated bounds, and then take the difference. Read more...

## Antiderivatives

The antiderivative of a function is a second function whose derivative is the first function. Read more...

## L’Hôpital’s Rule

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. Read more...

## Differentials and Approximation

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. Read more...

## Finding Extrema

Derivatives can be used to find a function’s local extreme values, its peaks and valleys. Read more...

## Derivatives of Non-Polynomial Functions

There are convenient rules the derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions. Read more...

## Properties of Derivatives

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? Read more...

## Chain Rule

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. Read more...

## Power Rule for Derivatives

There are some patterns that allow us to compute derivatives without having to compute the limit of the difference quotient. Read more...

## Derivatives and the Difference Quotient

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. Read more...

## Limits by Logarithms, Squeeze Theorem, and Euler’s Constant

Various tricks for evaluating tricky limits. Read more...

## Evaluating Limits

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. Read more...

## Compositions of Functions

Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function. Read more...

## Inverse Functions

Inverting a function entails reversing the outputs and inputs of the function. Read more...

## Reflections of Functions

When a function is reflected, it flips across one of the axes to become its mirror image. Read more...

## Rescalings of Functions

When a function is rescaled, it is stretched or compressed along one of the axes, like a slinky. Read more...

## Shifts of Functions

When a function is shifted, all of its points move vertically and/or horizontally by the same amount. Read more...

## Piecewise Functions

A piecewise function is pieced together from multiple different functions. Read more...

## Trigonometric Functions

Trigonometric functions represent the relationship between sides and angles in right triangles. Read more...

## Absolute Value

Absolute value represents the magnitude of a number, i.e. its distance from zero. Read more...

## Exponential and Logarithmic Functions

Exponential functions have variables as exponents. Logarithms cancel out exponentiation. Read more...

Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general. Read more...

## Graphing Rational Functions with Slant and Polynomial Asymptotes

A slant asymptote is a slanted line that arises from a linear term in the proper form of a rational function. Read more...

## Graphing Rational Functions with Horizontal and Vertical Asymptotes

If we choose one input on each side of an asymptote, we can tell which section of the plane the function will occupy. Read more...

## Vertical Asymptotes of Rational Functions

Vertical asymptotes are vertical lines that a function approaches but never quite reaches. Read more...

## Horizontal Asymptotes of Rational Functions

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. Read more...

## Polynomial Long Division

Polynomial long division works the same way as the long division algorithm that’s familiar from simple arithmetic. Read more...

## Sketching Graphs of Polynomials

We can sketch the graph of a polynomial using its end behavior and zeros. Read more...

## Rational Roots and Synthetic Division

The rational roots theorem can help us find zeros of polynomials without blindly guessing. Read more...

## Zeros of Polynomials

The zeros of a polynomial are the inputs that cause it to evaluate to zero. Read more...

## Standard Form and End Behavior of Polynomials

The end behavior of a polynomial refers to the type of output that is produced when we input extremely large positive or negative values. Read more...

## Systems of Inequalities

To solve a system of inequalities, we need to solve each individual inequality and find where all their solutions overlap. Read more...

## Linear Inequalities in the Plane

When a linear equation has two variables, the solution covers a section of the coordinate plane. Read more...

## Linear Inequalities in the Number Line

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. Read more...

## Vertex Form

To easily graph a quadratic equation, we can convert it to vertex form. Read more...

## Completing the Square

Completing the square helps us gain a better intuition for quadratic equations and understand where the quadratic formula comes from. Read more...

## Standard Form of a Quadratic Equation

Quadratic equations are similar to linear equations, except that they contain squares of a single variable. Read more...

## Linear Systems

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. Read more...

## Standard Form of a Line

Standard form makes it easy to see the intercepts of a line. Read more...

## Point-Slope Form

An easy way to write the equation of a line if we know the slope and a point on a line. Read more...

## Slope-Intercept Form

Introducing linear equations in two variables. Read more...

## Solving Linear Equations

Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division. Read more...

## Intuiting Ensemble Methods

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. Read more...

## Intuiting Decision Trees

Decision trees are able to model nonlinear data while remaining interpretable. Read more...

## Intuiting Neural Networks

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. Read more...

## Intuiting Support Vector Machines

A Support Vector Machine (SVM) computes the “best” separation between classes as the maximum-margin hyperplane. Read more...

## Intuiting Linear Regression

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). Read more...

## Intuiting Maximum a Posteriori and Maximum Likelihood Estimation

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. Read more...

## Intuiting Naive Bayes

Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. Read more...

## Applications of Calculus: Calculating the Horsepower of an Offensive Lineman

It comes out to roughly a fortieth of that of a truck. Read more...

## Applications of Calculus: Derivatives in String Art

String art works because the strings are tangent lines to a curve. Read more...

## Applications of Calculus: A Failure of Intuition

Calculus can show us how our intuition can fail us, a common theme in philosophy. Read more...

## History of Calculus: The Newton-Leibniz Controversy

Nobody came out of the dispute well. Read more...

## History of Calculus: The Man who “Broke” Math

When Joseph Fourier first introduced Fourier series, they gave mathematicians nightmares. Read more...

## Applications of Calculus: Continuously Compounded Interest

Deriving the “Pert” formula. Read more...

## Applications of Calculus: Maximizing Profit

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. Read more...

## Applications of Calculus: Optimization via Gradient Descent

Calculus can be used to find the parameters that minimize a function. Read more...

## Applications of Calculus: Physics Engines in Video Games

Physics engines use calculus to periodically updates the locations of objects. Read more...

## Applications of Calculus: Rendering 3D Computer Graphics

Introducing Kajiya’s rendering equation. Read more...

## Applications of Calculus: Rocket Propulsion

Deriving the ideal rocket equation. Read more...

## Applications of Calculus: Modeling Tumor Growth

Deriving the Gompertz function. Read more...

## Applications of Calculus: Understanding Plaque Buildup

Understanding why even slight narrowing of arteries can pose such a big problem to blood flow. Read more...

## Applications of Calculus: Cardiac Output

Measuring volume of blood the heart pumps out into the aorta per unit time. Read more...

## Intuiting Series

A series is the sum of a sequence. Read more...

## Intuiting Sequences

A sequence is a list of numbers that has some pattern. Read more...

## Intuiting Integrals

Integrals give the area under a portion of a function. Read more...

## Intuiting Derivatives

The derivative tells the steepness of a function at a given point, kind of like a carpenter’s level. Read more...

## Intuiting Limits

The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line. Read more...

## Intuiting Functions

A function is a scribble that crosses each vertical line only once. Read more...

## The Data Scientist’s Guide to Topological Data Analysis: Preamble

Bridging the communication gap between academia and industry in the field of TDA. Read more...

## Persistent Homology Software: Demonstration of TDA

Demonstrating an open-source implementation of persistent homology techniques in the TDA package for R. Read more...

## Intuiting Persistent Homology

Persistent homology provides a way to quantify the topological features that persist over our a data set’s full range of scale. Read more...

## Mapper Use-Cases at Aunalytics

At Aunalytics, Mapper outperformed hierarchical clustering in providing granular insights. Read more...

## Mapper Use-Cases at Ayasdi

Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data. Read more...

## Mapper Software: Demonstration of TDAmapper

Demonstrating an open-source implementation of Mapper in the TDAmapper package for R. Read more...

## Intuiting the Mapper Algorithm

Representing a data space’s topology by converting it into a network. Read more...

## A Game-Theoretic Analysis of Social Distancing During Epidemics

In a simplified problem framing, we investigate the (theoretical) usefulness of limiting the number of social connections per person. Read more...

## Making Indirect Interactions Explicit in Networks

Category theory provides a language for explicitly describing indirect relationships in graphs. Read more...

## Book Summary: Memory Evolutive Systems

Framing complex systems in the language of category theory. Read more...

## Introduction to Computers

The main ideas behind computers can be understood by anyone. Read more...

## The Brain in One Sentence

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. Read more...

## Shaping STDP Neural Networks with Periodic Stimulation: a Theoretical Analysis for the Case of Tree Networks

We solve a special case of how to periodically stimulate a neural network to obtain a desired connectivity. Read more...

## A Visual, Inductive Proof of Sharkovsky’s Theorem

Many existing proofs are not accessible to young mathematicians or those without experience in the realm of dynamic systems. Read more...

## The Physics Behind an Egg Drop: A Lively Story

Escaping a troll by using its own weight against itself. Read more...

## A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$

And a proof via double induction. Read more...

## Sound Waves

A brief overview of sound waves and how they interact with things. Read more...

## Detecting Dark Matter

A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP). Read more...

## Evidence for the Existence of Dark Matter

Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis’s impact on baryon density. Read more...