OpenStem

@openstem · Joined Jul 2026

7420 public items8 groups

Note~479 words · 2 min

Pick the shape before you start writing Most stuck proofs are stuck because the wrong technique was chosen in the first thirty seconds. The statement's form usually tells you which one to reach for. Direct proof Assume the hypothesis, walk

Mathematics

Mathematics · L4 · Mathematical proof techniques

@openstem

Note~424 words · 2 min

The most-used and most-misread number in statistics. Here's what it actually says. Definition The p-value is the probability of a test statistic at least this extreme, assuming H₀ is true. That z is a position on the curve below. Slide the

Mathematics

Mathematics · L4 · p-values and statistical significance: what they really mean

@openstem

Note~395 words · 2 min

One formula, and what each piece is called The denominator comes from the law of total probability: P(E) = ΣP(E|Hᵢ)P(Hᵢ). It's the normalising sum over every hypothesis you're considering. The medical test that catches everyone out Updating

Mathematics

Mathematics · L4 · Bayes' theorem: updating beliefs with evidence

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Note~385 words · 2 min

The right chart makes the finding obvious. The wrong one hides it, or worse, invents one. Start from your data type and your question, not from the chart you feel like drawing. Distribution of a single variable Relationship between two vari

Mathematics

Mathematics · L4 · Data visualisation: choosing the right chart

@openstem

Note~455 words · 2 min

A differential equation ties a function to its own rate of change. Separable equations An ODE is separable if it can be written . Move all the y-terms to one side and all the x-terms to the other, then integrate both sides independently: W

Mathematics

Mathematics · L4 · Solving first-order differential equations

@openstem

Note~392 words · 2 min

The derivative becomes a vector In one variable, the derivative is a number. In several, there's a rate of change in every direction, so "the derivative" becomes a whole vector of partials. Geometrically: slice the surface z = f(x,y) with t

Mathematics

Mathematics · L4 · Partial derivatives, gradients, and multivariable optimization

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Note~306 words · 2 min

Four axioms, and that's the entire definition A group is a set G with one binary operation ·. Four requirements, nothing else. Add commutativity (a·b = b·a) and the group is called abelian. Notice it isn't one of the four. The first example

Mathematics

Mathematics · L5 · Groups: Axioms and First Examples

@openstem

Note~349 words · 2 min

What a σ-algebra is for You can't consistently assign a length to every subset of ℝ. A σ-algebra is the collection of sets you agree to measure, chosen so the operations you need stay inside it. The pair (X, ℱ) is a measurable space. A meas

Mathematics

Mathematics · L5 · Measure & the Lebesgue Integral

@openstem

Note~376 words · 2 min

Probability is measure theory with one extra axiom A probability space is a triple (Ω, ℱ, P). Everything else is bookkeeping on top of it. That P(Ω) = 1 is the only difference from a general measure space. Everything from measure theory car

Mathematics

Mathematics · L5 · Measure-Theoretic Probability: Foundations

@openstem

Note~349 words · 2 min

The problem MCMC solves You know the posterior π(θ|data) only up to a normalising constant, because the denominator is an integral you can't do. Direct sampling is out. So build a Markov chain whose stationary distribution is π, run it, and

Mathematics

Mathematics · L5 · Bayesian Computation: MCMC

@openstem

Note~357 words · 2 min

Potential outcomes Every unit i has two outcomes waiting: Y_i(1) if treated, Y_i(0) if not. The individual effect is their difference. What you can recover is an average, because averages only need each unit to contribute one arm. Which ave

Mathematics

Mathematics · L5 · Causal Inference: Potential Outcomes

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Note~454 words · 2 min

Throwing away distance Topology asks what survives if you keep only the notion of "open" and discard distance entirely. The answer is nearly everything that matters: continuity, connectedness, compactness. The three axioms A topology τ on a

Mathematics

Mathematics · L5 · Point-Set Topology: Open Sets, Continuity, and Compactness

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Note~447 words · 2 min

One extra requirement, enormous consequences Complex differentiability demands the limit be the same no matter which direction h approaches 0 from. In ℝ there are only two directions. In ℂ there are infinitely many, and forcing them all to

Mathematics

Mathematics · L5 · Complex Analysis: Holomorphic Functions and Cauchy's Theorem

@openstem

Note~378 words · 2 min

The property that makes optimisation tractable Convexity is the line between problems you can solve and problems that are, in general, hopeless. In a convex problem the local view IS the global view, which is exactly what lets a gradient me

Mathematics

Mathematics · L5 · Convex Optimization: Convexity, Duality, and KKT Conditions

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Plot1 plot · 1 param

Mathematics

Mathematics · L2 · Proportions Are Straight Lines

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PlotPlot

Mathematics

Mathematics · L2 · Reading a Scatter Plot

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Plot1 plot · 2 params

Mathematics

Mathematics · L3 · Slope & Intercept: y = m·x + b

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Plot1 plot · 3 params

Mathematics

Mathematics · L3 · The Parabola: y = a·x² + b·x + c

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Plot3 segments

Mathematics

Mathematics · L3 · A Right Triangle on the Grid

@openstem

Plot1 plot · 2 params

Mathematics

Mathematics · L3 · The Sine Wave: y = a·sin(b·x)

@openstem

Plot1 plot · 2 params

Mathematics

Mathematics · L3 · The Cosine Wave: y = a·cos(b·x)

@openstem

Plot1 plot · 1 param

Mathematics

Mathematics · L3 · Exponential Growth: y = b^x

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Plot1 plot · 1 param

Mathematics

Mathematics · L3 · The Logarithm: y = a·ln(x)

@openstem

Plot1 plot · 2 params

Mathematics

Normal Distribution: exp(−(x−μ)²/(2σ²))

@openstem

Plot1 plot · 2 params

Mathematics

Linear Regression: y = m·x + b

@openstem

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Mathematics

Exponential Decay: y = e^(−k·x)

@openstem

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Mathematics

CDF Growth: y = 1 − e^(−k·x)

@openstem

Plot1 plot · 1 param

Mathematics

Quadratic Fit: y = a·x²

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Plot1 plot · 2 params

Mathematics

Mathematics · L5 · Normal Density: shift the mean μ and scale σ

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Flowchart15 objects

Mathematics

Mathematics · L3 · Absolute Value: Equation vs '<' vs '>'

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