{ "version": "https://jsonfeed.org/version/1.1", "title": "Lemma — Today's Pick", "home_page_url": "https://lemma.wiki", "feed_url": "https://lemma.wiki/feed.json", "language": "en-US", "description": "One application, one module, one shape, one journey — rotated daily.", "items": [ { "id": "lemma:20587:application:bitcoin-signature", "url": "https://lemma.wiki/en/finance/bitcoin-signature", "title": "finance · The Bitcoin Signature", "content_text": "Anyone can see your Bitcoin address. No one should see your private key. So how can the network verify that you authorized a payment without learning the secret that authorizes all payments?", "date_published": "2026-05-14T00:00:00.000Z", "tags": [ "application", "finance" ] }, { "id": "lemma:20587:module:distributions", "url": "https://lemma.wiki/en/modules/distributions", "title": "module · Distributions", "content_text": "A probability is one guess. A distribution is the whole shape of uncertainty. The thing softmax produces, the thing a histogram is, the thing a return is drawn from. Most quantities a model predicts or a portfolio holds are distributions before they are numbers.", "date_published": "2026-05-14T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20587:shape:equilibrium", "url": "https://lemma.wiki/en/shapes/equilibrium", "title": "shape · Equilibrium", "content_text": "Two opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.", "date_published": "2026-05-14T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20587:journey:where-change-vanishes", "url": "https://lemma.wiki/en/journey/where-change-vanishes", "title": "journey · 4 days · Where Change Vanishes", "content_text": "One question — *where does change vanish?* — under three physics applications.", "date_published": "2026-05-14T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20586:application:portfolio-risk", "url": "https://lemma.wiki/en/finance/portfolio-risk", "title": "finance · Portfolio Risk", "content_text": "Two risky assets, mixed, can be less risky than either alone. The mathematics is one cross-term in a quadratic — and it's the same identity that makes a vector sum's length more (or less) than the sum of its parts. Variance, covariance, correlation; the dot product of returns.", "date_published": "2026-05-13T00:00:00.000Z", "tags": [ "application", "finance" ] }, { "id": "lemma:20586:module:entropy", "url": "https://lemma.wiki/en/modules/entropy", "title": "module · Entropy", "content_text": "20 questions to find any one of N items needs log₂ N when items are equally likely. Entropy is the generalization — the expected number of yes/no questions when they aren't. The bound everything from Wordle to Huffman to password strength bumps against.", "date_published": "2026-05-13T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20586:shape:find-the-minimum", "url": "https://lemma.wiki/en/shapes/find-the-minimum", "title": "shape · Find the Minimum", "content_text": "The nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.", "date_published": "2026-05-13T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20586:journey:how-compression-works", "url": "https://lemma.wiki/en/journey/how-compression-works", "title": "journey · 4 days · How Compression Works", "content_text": "One three-step procedure under JPEG, TF-IDF, and the raw-pixel entropy floor — across graphics and ML.", "date_published": "2026-05-13T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20585:application:bezier-curves", "url": "https://lemma.wiki/en/graphics/bezier-curves", "title": "graphics · Bezier Curves", "content_text": "A designer drags four handles. A movie character gets a smooth cheek, a car gets a perfect hood, a letter gets its curve. How does a computer turn a few points into a smooth path?", "date_published": "2026-05-12T00:00:00.000Z", "tags": [ "application", "graphics" ] }, { "id": "lemma:20585:module:vectors", "url": "https://lemma.wiki/en/modules/vectors", "title": "module · Vectors", "content_text": "A point says where. A vector says how to move. The same tuple plays four roles — position, displacement, velocity, feature — across graphics, physics, and ML. Two operations (add and scale) carry every one of them.", "date_published": "2026-05-12T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20585:shape:compression", "url": "https://lemma.wiki/en/shapes/compression", "title": "shape · Compression", "content_text": "High-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.", "date_published": "2026-05-12T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20585:journey:vectors-everywhere", "url": "https://lemma.wiki/en/journey/vectors-everywhere", "title": "journey · 5 days · Vectors Everywhere", "content_text": "Same tuple, four roles — across graphics, physics, ML, and finance.", "date_published": "2026-05-12T00:00:00.000Z", "tags": [ "journey", "journey · 5 days" ] }, { "id": "lemma:20584:application:image-compression", "url": "https://lemma.wiki/en/graphics/image-compression", "title": "graphics · Why Images Compress", "content_text": "Two same-size images: one shrinks, one doesn't. The histogram tells half the story; the spatial structure tells the rest. The entropy module names the floor — this page shows what that floor isn't bounding.", "date_published": "2026-05-11T00:00:00.000Z", "tags": [ "application", "graphics" ] }, { "id": "lemma:20584:module:linearization", "url": "https://lemma.wiki/en/modules/linearization", "title": "module · Linearization", "content_text": "Most equations are hard. Their tangent line at a point is easy. Replace one with the other and you get a tool that powers the pendulum clock, Newton's method, and gradient descent — valid in a regime, wrong outside it. The discipline is the regime.", "date_published": "2026-05-11T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20584:shape:inverse-problem", "url": "https://lemma.wiki/en/shapes/inverse-problem", "title": "shape · The Inverse Problem", "content_text": "The forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.", "date_published": "2026-05-11T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20584:journey:finding-the-minimum", "url": "https://lemma.wiki/en/journey/finding-the-minimum", "title": "journey · 4 days · Finding the Minimum", "content_text": "One skeleton — objective, move, step, stop — under three applications across ML and finance.", "date_published": "2026-05-11T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20583:application:jpeg-compression", "url": "https://lemma.wiki/en/graphics/jpeg-compression", "title": "graphics · Why JPEG Throws Pixels Away", "content_text": "Lossy compression isn't bound by entropy — it picks what to discard. JPEG changes basis (DCT) so the picture becomes sparse, throws away the coordinates that don't matter (quantization), and Huffman-packs the rest. Three steps, one savings: fewer coefficients, smaller values, longer zero runs.", "date_published": "2026-05-10T00:00:00.000Z", "tags": [ "application", "graphics" ] }, { "id": "lemma:20583:module:integration", "url": "https://lemma.wiki/en/modules/integration", "title": "module · The Integral", "content_text": "A speedometer reads. How far have you traveled? The integral is what survives when you sum a rate over time. The pair to the derivative — and the fundamental theorem says they are inverses.", "date_published": "2026-05-10T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20583:shape:equilibrium", "url": "https://lemma.wiki/en/shapes/equilibrium", "title": "shape · Equilibrium", "content_text": "Two opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.", "date_published": "2026-05-10T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20583:journey:change-and-accumulation", "url": "https://lemma.wiki/en/journey/change-and-accumulation", "title": "journey · 7 days · Change and accumulation in 7 days", "content_text": "How calculus runs both physics and finance — same math, different nouns.", "date_published": "2026-05-10T00:00:00.000Z", "tags": [ "journey", "journey · 7 days" ] }, { "id": "lemma:20582:application:curve-intersections", "url": "https://lemma.wiki/en/graphics/curve-intersections", "title": "graphics · Curve Intersections", "content_text": "Drag two conics. Sometimes you see four crossings. Sometimes two. Sometimes none. Bezout says the count is still four. Where did the missing intersections go — and what does a graphics engine do about it?", "date_published": "2026-05-09T00:00:00.000Z", "tags": [ "application", "graphics" ] }, { "id": "lemma:20582:module:bezout", "url": "https://lemma.wiki/en/modules/bezout", "title": "module · Bezout's Theorem", "content_text": "Two curves of degrees d, e meet in exactly d·e points — once the plane is repaired three ways. The chord-and-tangent feeds elliptic-curve arithmetic, which feeds Bitcoin signatures.", "date_published": "2026-05-09T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20582:shape:find-the-minimum", "url": "https://lemma.wiki/en/shapes/find-the-minimum", "title": "shape · Find the Minimum", "content_text": "The nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.", "date_published": "2026-05-09T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20582:journey:to-backprop", "url": "https://lemma.wiki/en/journey/to-backprop", "title": "journey · 7 days · To Backprop in 7 days", "content_text": "Why does a model train? It walks downhill. The hill is a function.", "date_published": "2026-05-09T00:00:00.000Z", "tags": [ "journey", "journey · 7 days" ] }, { "id": "lemma:20581:application:projectile-motion", "url": "https://lemma.wiki/en/physics/projectile-motion", "title": "physics · Projectile Motion", "content_text": "Throw a ball. Ignore air. Its horizontal motion keeps time; its vertical motion loses to gravity. Why does that make a parabola?", "date_published": "2026-05-08T00:00:00.000Z", "tags": [ "application", "physics" ] }, { "id": "lemma:20581:module:derivatives", "url": "https://lemma.wiki/en/modules/derivatives", "title": "module · The Derivative", "content_text": "A moving point leaves a trail. The derivative is not the trail — it is the arrow the trail wants to become at this instant. Secant slopes converge to tangent slopes, and the same machine becomes slope, velocity, and rate.", "date_published": "2026-05-08T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20581:shape:compression", "url": "https://lemma.wiki/en/shapes/compression", "title": "shape · Compression", "content_text": "High-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.", "date_published": "2026-05-08T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20581:journey:to-bitcoin", "url": "https://lemma.wiki/en/journey/to-bitcoin", "title": "journey · 7 days · To Bitcoin in 7 days", "content_text": "From the most expensive pizza in history to the signature that keeps a wallet safe.", "date_published": "2026-05-08T00:00:00.000Z", "tags": [ "journey", "journey · 7 days" ] }, { "id": "lemma:20580:application:terminal-velocity", "url": "https://lemma.wiki/en/physics/terminal-velocity", "title": "physics · Why Falling Stops Speeding Up", "content_text": "Gravity pulls forever, but a falling raindrop doesn't speed up forever. The reason is one first-order equation — a derivative balanced against a force that grows with speed. Terminal velocity is not a maximum; it's an equilibrium.", "date_published": "2026-05-07T00:00:00.000Z", "tags": [ "application", "physics" ] }, { "id": "lemma:20580:module:parametric-curves", "url": "https://lemma.wiki/en/modules/parametric-curves", "title": "module · Parametric Curves", "content_text": "A curve is not a picture. Three motions can paint the same parabola — same image, different parametrizations. Pin down the distinction the word 'curve' was hiding, and a whole stack of downstream tools snaps into place.", "date_published": "2026-05-07T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20580:shape:inverse-problem", "url": "https://lemma.wiki/en/shapes/inverse-problem", "title": "shape · The Inverse Problem", "content_text": "The forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.", "date_published": "2026-05-07T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20580:journey:working-backward", "url": "https://lemma.wiki/en/journey/working-backward", "title": "journey · 4 days · Working Backward", "content_text": "Three pages, one question — given the output, find the input.", "date_published": "2026-05-07T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20579:application:pendulum-clock", "url": "https://lemma.wiki/en/physics/pendulum-clock", "title": "physics · The Pendulum Clock", "content_text": "Double the swing. Why does the clock barely change? A real pendulum's period depends on amplitude — but for small swings it almost doesn't. Linearize sin θ ≈ θ, and the period becomes constant. The clock stands on that approximation.", "date_published": "2026-05-06T00:00:00.000Z", "tags": [ "application", "physics" ] }, { "id": "lemma:20579:module:log", "url": "https://lemma.wiki/en/modules/log", "title": "module · The Logarithm", "content_text": "The trick that turns × into +. The whole module is one equation; everything else is consequence.", "date_published": "2026-05-06T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20579:shape:equilibrium", "url": "https://lemma.wiki/en/shapes/equilibrium", "title": "shape · Equilibrium", "content_text": "Two opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.", "date_published": "2026-05-06T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20579:journey:where-change-vanishes", "url": "https://lemma.wiki/en/journey/where-change-vanishes", "title": "journey · 4 days · Where Change Vanishes", "content_text": "One question — *where does change vanish?* — under three physics applications.", "date_published": "2026-05-06T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20578:application:damped-oscillator", "url": "https://lemma.wiki/en/physics/damped-oscillator", "title": "physics · Why Things Stop Swinging", "content_text": "A pendulum left alone slows and stops; the same pendulum pushed on the beat swings higher and higher. One equation — ẍ + 2γ·ẋ + ω₀²·x = F(t) — runs both. Damping decides how it stops; resonance decides when an outside push wins.", "date_published": "2026-05-05T00:00:00.000Z", "tags": [ "application", "physics" ] }, { "id": "lemma:20578:module:optimization", "url": "https://lemma.wiki/en/modules/optimization", "title": "module · Optimization", "content_text": "Optimization is not finding the formula. It is choosing a quantity to improve, then moving through possible choices until improvement stops. The same five steps — objective, search space, move, step size, stopping — run gradient descent in ML, weight selection in portfolios, and temperature fitting in calibration. Different algorithms, identical bones.", "date_published": "2026-05-05T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20578:shape:find-the-minimum", "url": "https://lemma.wiki/en/shapes/find-the-minimum", "title": "shape · Find the Minimum", "content_text": "The nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.", "date_published": "2026-05-05T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20578:journey:how-compression-works", "url": "https://lemma.wiki/en/journey/how-compression-works", "title": "journey · 4 days · How Compression Works", "content_text": "One three-step procedure under JPEG, TF-IDF, and the raw-pixel entropy floor — across graphics and ML.", "date_published": "2026-05-05T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20577:application:bitcoin-pizza", "url": "https://lemma.wiki/en/finance/bitcoin-pizza", "title": "finance · The Bitcoin Pizza", "content_text": "On May 22, 2010, a Florida programmer paid 10,000 BTC for two Papa John's pizzas — about $41. Sixteen years later, those coins are worth $1 billion. The most expensive meal in history.", "date_published": "2026-05-04T00:00:00.000Z", "tags": [ "application", "finance" ] }, { "id": "lemma:20577:module:distributions", "url": "https://lemma.wiki/en/modules/distributions", "title": "module · Distributions", "content_text": "A probability is one guess. A distribution is the whole shape of uncertainty. The thing softmax produces, the thing a histogram is, the thing a return is drawn from. Most quantities a model predicts or a portfolio holds are distributions before they are numbers.", "date_published": "2026-05-04T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20577:shape:compression", "url": "https://lemma.wiki/en/shapes/compression", "title": "shape · Compression", "content_text": "High-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.", "date_published": "2026-05-04T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20577:journey:vectors-everywhere", "url": "https://lemma.wiki/en/journey/vectors-everywhere", "title": "journey · 5 days · Vectors Everywhere", "content_text": "Same tuple, four roles — across graphics, physics, ML, and finance.", "date_published": "2026-05-04T00:00:00.000Z", "tags": [ "journey", "journey · 5 days" ] }, { "id": "lemma:20576:application:present-value", "url": "https://lemma.wiki/en/finance/present-value", "title": "finance · What Is Future Money Worth Today?", "content_text": "A future dollar is not a present dollar. Discounting and integration price every cash-flow stream — from rent to bonds to perpetuities — through one identity: future cash times the discount factor, summed across time.", "date_published": "2026-05-03T00:00:00.000Z", "tags": [ "application", "finance" ] }, { "id": "lemma:20576:module:entropy", "url": "https://lemma.wiki/en/modules/entropy", "title": "module · Entropy", "content_text": "20 questions to find any one of N items needs log₂ N when items are equally likely. Entropy is the generalization — the expected number of yes/no questions when they aren't. The bound everything from Wordle to Huffman to password strength bumps against.", "date_published": "2026-05-03T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20576:shape:inverse-problem", "url": "https://lemma.wiki/en/shapes/inverse-problem", "title": "shape · The Inverse Problem", "content_text": "The forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.", "date_published": "2026-05-03T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20576:journey:finding-the-minimum", "url": "https://lemma.wiki/en/journey/finding-the-minimum", "title": "journey · 4 days · Finding the Minimum", "content_text": "One skeleton — objective, move, step, stop — under three applications across ML and finance.", "date_published": "2026-05-03T00:00:00.000Z", "tags": [ "journey", "journey · 4 days" ] }, { "id": "lemma:20575:application:tf-idf", "url": "https://lemma.wiki/en/ml/tf-idf", "title": "ml / dl · TF-IDF", "content_text": "Why does Google show those results in that order? For thirty years the decision rule barely changed. TF-IDF ranks documents by a score that looks like a sum of probabilities — and the log inside it is the same log as in entropy. Rare words carry many bits; common words carry near zero. Stopwords aren't a list, they're the zero set of a function.", "date_published": "2026-05-02T00:00:00.000Z", "tags": [ "application", "ml / dl" ] }, { "id": "lemma:20575:module:vectors", "url": "https://lemma.wiki/en/modules/vectors", "title": "module · Vectors", "content_text": "A point says where. A vector says how to move. The same tuple plays four roles — position, displacement, velocity, feature — across graphics, physics, and ML. Two operations (add and scale) carry every one of them.", "date_published": "2026-05-02T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20575:shape:equilibrium", "url": "https://lemma.wiki/en/shapes/equilibrium", "title": "shape · Equilibrium", "content_text": "Two opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.", "date_published": "2026-05-02T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20575:journey:change-and-accumulation", "url": "https://lemma.wiki/en/journey/change-and-accumulation", "title": "journey · 7 days · Change and accumulation in 7 days", "content_text": "How calculus runs both physics and finance — same math, different nouns.", "date_published": "2026-05-02T00:00:00.000Z", "tags": [ "journey", "journey · 7 days" ] }, { "id": "lemma:20574:application:model-calibration", "url": "https://lemma.wiki/en/ml/model-calibration", "title": "ml / dl · Model Calibration", "content_text": "A model says '70% confident.' Does that mean, across many such predictions, seven in ten are right? The number on the screen and the long-run frequency are two different quantities. The reliability diagram makes the gap visible; one scalar — temperature — rotates the curve back to the diagonal.", "date_published": "2026-05-01T00:00:00.000Z", "tags": [ "application", "ml / dl" ] }, { "id": "lemma:20574:module:linearization", "url": "https://lemma.wiki/en/modules/linearization", "title": "module · Linearization", "content_text": "Most equations are hard. Their tangent line at a point is easy. Replace one with the other and you get a tool that powers the pendulum clock, Newton's method, and gradient descent — valid in a regime, wrong outside it. The discipline is the regime.", "date_published": "2026-05-01T00:00:00.000Z", "tags": [ "module", "module" ] }, { "id": "lemma:20574:shape:find-the-minimum", "url": "https://lemma.wiki/en/shapes/find-the-minimum", "title": "shape · Find the Minimum", "content_text": "The nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.", "date_published": "2026-05-01T00:00:00.000Z", "tags": [ "shape", "shape" ] }, { "id": "lemma:20574:journey:to-backprop", "url": "https://lemma.wiki/en/journey/to-backprop", "title": "journey · 7 days · To Backprop in 7 days", "content_text": "Why does a model train? It walks downhill. The hill is a function.", "date_published": "2026-05-01T00:00:00.000Z", "tags": [ "journey", "journey · 7 days" ] } ] }