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Book Chapters

Decision Trees

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

K-Means Clustering

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

Single-Variable Gradient Descent

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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Coding

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Decision Trees

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

K-Means Clustering

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

Single-Variable Gradient Descent

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...

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Algebra

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...

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...

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...

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Calculus

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...

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...

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...

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...

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...

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...

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...

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...

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Blog

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

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. Read more...

For Most Students, Competition Math is a Waste of Time

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. 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...

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Q&A (Misc)

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Machine Learning

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

Decision Trees

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

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 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 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 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...

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Q&A

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

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Teaching

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

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. Read more...

For Most Students, Competition Math is a Waste of Time

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. Read more...

Back to Top ↑

Linear Algebra

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...

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...

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...

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Graphs

Decision Trees

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

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Applications

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Graphing Calculator

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Drawing

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Algorithms

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 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 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 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...

Back to Top ↑

Limits and Derivatives

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...

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...

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...

Back to Top ↑

Objects

Euler Estimation

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

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Regression

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Stories

Business Lessons from Science Fair

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. Read more...

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Research

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Topological Data Analysis

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...

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History

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Differential Equations

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...

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...

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...

Back to Top ↑

Neural Networks

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

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Blog (Tier 2)

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Business Lessons from Science Fair

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. Read more...

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Physics

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Integrals

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...

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Quadratic Equations

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...

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Series

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Artificial Intelligence

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Games

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Mapper

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...

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Linear Equations and Systems

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...

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Rational Functions

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Non-Polynomial Functions

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Transformations of Functions

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...

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Vectors

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...

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Matrices

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Simulation

Euler Estimation

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

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Searching

Single-Variable Gradient Descent

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...

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Notation

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Arithmetic

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Limits

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...

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Inequalities

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Polynomials

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Python

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Volume

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...

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Eigenspace

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...

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Hello World

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Computer Science

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Neuroevolution

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Blondie24

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Quant

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Strength Training

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Proofs

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Learning

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Persistent Homology

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Videos

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Geometry

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Math Academy

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Eurisko

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Classification

Decision Trees

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

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Tips

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Calisthenics

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Student Errors

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Education

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Expository

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Neuroscience

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...

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Category Theory

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Functions

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Sequences

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Blog (Tier 1)

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Multivariable Calculus

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Sorting

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...

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Game Trees

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Gymnastic Rings

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Gifted Students

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Educational Acceleration

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Science Fair

Business Lessons from Science Fair

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. Read more...

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Grading

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Intelligence

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

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. Read more...

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Logic

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Technical Diary

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

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Humanities

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Computers

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Book Summaries

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Game Theory

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Homology

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Derivatives

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LaTeX

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Linear Programming

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Archetypes

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Tensors

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Personal Website

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College Applications

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Syllabus

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Differentials

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Absolute Value

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Terminology

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Education Policy

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Math Wars

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AI

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Learning Strategies

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Riddles

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Career

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Probability

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Competition Math

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Moore Method

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Project-Based Learning

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Datasets

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Bayesian Statistics

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