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The area under the bell curve has no formula. Mathematicians proved that in 1830 — not that nobody found it yet, that it literally cannot exist. Then someone squared the integral, switched to polar coordinates, and got √π in four minutes. The same curve LLMs assume for weight noise...

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Deliriumm

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31,467 Aufrufe • vor 11 Tagen •via X (Twitter)

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Profilbild von вяεzια тσмsση
вяεzια тσмsσηvor 10 Tagen

Mechanically. Fill it with h2o. Place blotter on the water. Measure wet spot.

Profilbild von SPORTS EVANGELIST IS NOT SPORTS ILLUSTRATED
SPORTS EVANGELIST IS NOT SPORTS ILLUSTRATEDvor 11 Tagen

@grok please

Profilbild von milorad
miloradvor 10 Tagen

Do you have a YouTube channel?

Profilbild von Paul Bogaars
Paul Bogaarsvor 11 Tagen

The Bell curve and the Ricci Flow curve are One of the Same !!! Just at different times or path of their trajectory !!!

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In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
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In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

23,254 Aufrufe • vor 4 Monaten

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
0:29

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

106,981 Aufrufe • vor 9 Monaten

MIT filmed a mathematician proving the theorem behind every equation Maxwell ever wrote. Most engineers use it daily without understanding why any surface gives the same answer - and the ones who can explain it earn $350K at aerospace firms. Stokes' theorem is the reason Maxwell could write four equations instead of forty. Every result in electromagnetism, fluid mechanics, and differential geometry sits on top of it. Understanding it changes how you see every circulation and flux problem in physics. This is Denis Auroux. MIT, 18.02, Multivariable Calculus, Fall 2007. Lecture 31. It covers Stokes' theorem - the result that unifies line integrals, surface integrals, and curl into one statement. He opens with one confession. Then the statement. The work done by a vector field along any closed curve equals the flux of its curl through any surface bounded by that curve. Not a specific surface. Any surface you want. You pick whichever one makes the calculation easier. Then Green's theorem disappears. The result from the plane that converts line integrals to double integrals turns out to be Stokes' theorem in disguise. Same formula, restricted to a flat surface in the xy-plane. What looked like two separate theorems is one. Then orientation. Walk along the curve with the surface to your left and the normal vector points up. This rule is not chosen for elegance. It is the price of having a consistent coordinate system in three dimensions. Then the proof. Cut your surface into tiny flat tiles. Apply Green's theorem to each one. Sum everything up. All interior boundary edges cancel. Only the outer edge survives. The sum of tiny Green's theorems gives you Stokes' theorem for any surface. Watch the moment he computes the same line integral two ways. First directly - answer is pi. Then through a paraboloid instead of the obvious flat disk. Longer calculation, stranger setup, same answer. The theorem is not an abstraction. You can watch it work. A fluid dynamics engineer I know kept this lecture open during their first week modeling atmospheric circulation. Said it was the first time Stokes felt like a tool rather than a formula to verify on exams. Free on YouTube, MIT OpenCourseWare, 18.02. bookmark this and watch later - after this lecture every circulation, every flux, and every curl you encounter will feel like three views of the same object
47:56

MIT filmed a mathematician proving the theorem behind every equation Maxwell ever wrote. Most engineers use it daily without understanding why any surface gives the same answer - and the ones who can explain it earn $350K at aerospace firms. Stokes' theorem is the reason Maxwell could write four equations instead of forty. Every result in electromagnetism, fluid mechanics, and differential geometry sits on top of it. Understanding it changes how you see every circulation and flux problem in physics. This is Denis Auroux. MIT, 18.02, Multivariable Calculus, Fall 2007. Lecture 31. It covers Stokes' theorem - the result that unifies line integrals, surface integrals, and curl into one statement. He opens with one confession. Then the statement. The work done by a vector field along any closed curve equals the flux of its curl through any surface bounded by that curve. Not a specific surface. Any surface you want. You pick whichever one makes the calculation easier. Then Green's theorem disappears. The result from the plane that converts line integrals to double integrals turns out to be Stokes' theorem in disguise. Same formula, restricted to a flat surface in the xy-plane. What looked like two separate theorems is one. Then orientation. Walk along the curve with the surface to your left and the normal vector points up. This rule is not chosen for elegance. It is the price of having a consistent coordinate system in three dimensions. Then the proof. Cut your surface into tiny flat tiles. Apply Green's theorem to each one. Sum everything up. All interior boundary edges cancel. Only the outer edge survives. The sum of tiny Green's theorems gives you Stokes' theorem for any surface. Watch the moment he computes the same line integral two ways. First directly - answer is pi. Then through a paraboloid instead of the obvious flat disk. Longer calculation, stranger setup, same answer. The theorem is not an abstraction. You can watch it work. A fluid dynamics engineer I know kept this lecture open during their first week modeling atmospheric circulation. Said it was the first time Stokes felt like a tool rather than a formula to verify on exams. Free on YouTube, MIT OpenCourseWare, 18.02. bookmark this and watch later - after this lecture every circulation, every flux, and every curl you encounter will feel like three views of the same object

Zyphor

11,286 Aufrufe • vor 1 Monat

Lecture 4 on Calculus of Variations You might wonder...If I’m optimizing a shape...a curve, a surface, a whole path, what does "take the derivative and set it to zero" even mean? Do I take the damn derivative with respect to a curve/surface? 🤔 In normal calculus the variable is a number x, so the reflex is clean...f′(x)=0. In calculus of variations the variable is a whole function...the geometry itself, like a curve y(x) (or a surface z(x,y)). So the derivative can’t be a single slope. It has to be a pointwise sensitivity, i.e. how the objective reacts to tiny local deformations. You’re holding a whole shape, like a curve y(x). Your objective isn’t f(x) anymore, it’s a functional J[y], and as we've seen with our first there examples, usually an integral that depends on the entire curve (often through y and y’). To talk about a “derivative”, you do the only thing that makes sense: you nudge the entire curve by a tiny amount and see how J changes. Pick a wiggle shape η(x). It’s not random...it’s any admissible deformation direction. Admissible just means it obeys the constraints. If the endpoints are fixed, you force η(0)=η(1)=0 so the wiggle doesn’t move the endpoints. Then scale that wiggle by a small number ε and define the perturbed curve yε(x)=y(x)+εη(x). Now treat ε like the usual scalar in a Taylor expansion. As ε→0, J[y+εη] expands as J[y+εη] = J[y] + ε · (first-order term depending linearly on η) + o(ε). So the difference is J[y+εη] - J[y] = ε · (linear functional of η) + o(ε). For the standard integral of a Lagrangian problems, that linear functional can be written as an inner product with some function of x: J[y+εη] - J[y] = ε ∫ (δJ/δy)(x) η(x) dx + o(ε). That’s the definition-level meaning of δJ/δy: it’s the unique pointwise sensitivity function that makes this identity true for every admissible η. If δJ/δy is positive at some x, then choosing η negative there decreases J; if δJ/δy is negative there, pushing y upward locally decreases J. It’s literally a map along the curve saying push this way to go downhill. Now translate “set the derivative to zero.” At a minimizer y*, the first-order change must vanish for every admissible wiggle: J[y*+εη] − J[y*] = o(ε) for all η. Plug in the expansion and the ε-term must be zero: ∫ (δJ/δy)(x) η(x) dx = 0 for all admissible η. Here’s the crucial logic step: the only way an integral against every test function η can be zero is if the integrand itself is zero (in the usual sense used in analysis). So you get δJ/δy = 0. For the common case J[y]=∫ L(x, y, y’) dx, you can compute δJ/δy explicitly and it becomes the Euler–Lagrange expression δJ/δy = ∂L/∂y − d/dx(∂L/∂y’). So if you name the Euler–Lagrange residual as “left-hand side” R(x) = ∂L/∂y − d/dx(∂L/∂y’), then “set the derivative to zero” is exactly R(x)=0. That’s why animation works so well. You don’t have to solve R=0 in one shot. You can evolve the curve in an artificial time τ by pushing it in the downhill direction: ∂y/∂τ = −R(y). Where the residual is large, the curve moves a lot; as the residual drains toward zero, the motion dies out and the curve settles into an extremal. In our animations, we start from an intentionally ugly curve/surface. Frame by frame the functional drops, the residual drains away, and the geometry relaxes into an extremal. #CalculusOfVariations #EulerLagrange #FunctionalDerivative #GradientFlow #Optimization #MathAnimation
0:58

Lecture 4 on Calculus of Variations You might wonder...If I’m optimizing a shape...a curve, a surface, a whole path, what does "take the derivative and set it to zero" even mean? Do I take the damn derivative with respect to a curve/surface? 🤔 In normal calculus the variable is a number x, so the reflex is clean...f′(x)=0. In calculus of variations the variable is a whole function...the geometry itself, like a curve y(x) (or a surface z(x,y)). So the derivative can’t be a single slope. It has to be a pointwise sensitivity, i.e. how the objective reacts to tiny local deformations. You’re holding a whole shape, like a curve y(x). Your objective isn’t f(x) anymore, it’s a functional J[y], and as we've seen with our first there examples, usually an integral that depends on the entire curve (often through y and y’). To talk about a “derivative”, you do the only thing that makes sense: you nudge the entire curve by a tiny amount and see how J changes. Pick a wiggle shape η(x). It’s not random...it’s any admissible deformation direction. Admissible just means it obeys the constraints. If the endpoints are fixed, you force η(0)=η(1)=0 so the wiggle doesn’t move the endpoints. Then scale that wiggle by a small number ε and define the perturbed curve yε(x)=y(x)+εη(x). Now treat ε like the usual scalar in a Taylor expansion. As ε→0, J[y+εη] expands as J[y+εη] = J[y] + ε · (first-order term depending linearly on η) + o(ε). So the difference is J[y+εη] - J[y] = ε · (linear functional of η) + o(ε). For the standard integral of a Lagrangian problems, that linear functional can be written as an inner product with some function of x: J[y+εη] - J[y] = ε ∫ (δJ/δy)(x) η(x) dx + o(ε). That’s the definition-level meaning of δJ/δy: it’s the unique pointwise sensitivity function that makes this identity true for every admissible η. If δJ/δy is positive at some x, then choosing η negative there decreases J; if δJ/δy is negative there, pushing y upward locally decreases J. It’s literally a map along the curve saying push this way to go downhill. Now translate “set the derivative to zero.” At a minimizer y*, the first-order change must vanish for every admissible wiggle: J[y*+εη] − J[y*] = o(ε) for all η. Plug in the expansion and the ε-term must be zero: ∫ (δJ/δy)(x) η(x) dx = 0 for all admissible η. Here’s the crucial logic step: the only way an integral against every test function η can be zero is if the integrand itself is zero (in the usual sense used in analysis). So you get δJ/δy = 0. For the common case J[y]=∫ L(x, y, y’) dx, you can compute δJ/δy explicitly and it becomes the Euler–Lagrange expression δJ/δy = ∂L/∂y − d/dx(∂L/∂y’). So if you name the Euler–Lagrange residual as “left-hand side” R(x) = ∂L/∂y − d/dx(∂L/∂y’), then “set the derivative to zero” is exactly R(x)=0. That’s why animation works so well. You don’t have to solve R=0 in one shot. You can evolve the curve in an artificial time τ by pushing it in the downhill direction: ∂y/∂τ = −R(y). Where the residual is large, the curve moves a lot; as the residual drains toward zero, the motion dies out and the curve settles into an extremal. In our animations, we start from an intentionally ugly curve/surface. Frame by frame the functional drops, the residual drains away, and the geometry relaxes into an extremal. #CalculusOfVariations #EulerLagrange #FunctionalDerivative #GradientFlow #Optimization #MathAnimation

Mathelirium

12,186 Aufrufe • vor 9 Monaten

In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth into the sine and cosine waves that live inside every transformer attention layer. The proof runs through Briggs' 1617 zooming algorithm, complex polar coordinates, and a thin-triangle argument that connects imaginary exponents to angles in radians. When you plug I into the exponential function, you don't get growth - you get rotation confined to the unit circle. The same unit circle that softmax maps every attention score onto before the GPU multiplies them. 27 minutes. Bookmark & watch today. The most beautiful equation in mathematics and why it powers every model you have ever used.
26:53

In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth into the sine and cosine waves that live inside every transformer attention layer. The proof runs through Briggs' 1617 zooming algorithm, complex polar coordinates, and a thin-triangle argument that connects imaginary exponents to angles in radians. When you plug I into the exponential function, you don't get growth - you get rotation confined to the unit circle. The same unit circle that softmax maps every attention score onto before the GPU multiplies them. 27 minutes. Bookmark & watch today. The most beautiful equation in mathematics and why it powers every model you have ever used.

Deliriumm

118,141 Aufrufe • vor 1 Tag

Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.
15:45

Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.

Eron

18,166 Aufrufe • vor 1 Monat

For 300 years mathematicians solved polynomials of degree 1, 2, 3, and 4 with increasingly complex formulas. Then they hit degree 5 and the formula simply doesn't exist - not undiscovered, provably impossible. The reason is group theory. The 5 solutions of a quintic can be scrambled in ways that no finite chain of nested roots can replicate - the same symmetry mathematics that underlies modern cryptography and the attention mechanism in transformers. 28 minutes. Bookmark & watch today. The proof is more beautiful than the formula would have been.
27:00

For 300 years mathematicians solved polynomials of degree 1, 2, 3, and 4 with increasingly complex formulas. Then they hit degree 5 and the formula simply doesn't exist - not undiscovered, provably impossible. The reason is group theory. The 5 solutions of a quintic can be scrambled in ways that no finite chain of nested roots can replicate - the same symmetry mathematics that underlies modern cryptography and the attention mechanism in transformers. 28 minutes. Bookmark & watch today. The proof is more beautiful than the formula would have been.

Eron

34,543 Aufrufe • vor 1 Monat

There's a profound elegance in watching gradient descent find the signal in the noise. This is a simple MLP with sigmoid activations slowly warping to approximate f(x) = 1/(1 + x²). Watch the loss curve right around epoch 400. That sudden phase transition where the network finally "gets it" and snaps into the Witch of Agnesi shape is incredibly satisfying to watch.
0:20

There's a profound elegance in watching gradient descent find the signal in the noise. This is a simple MLP with sigmoid activations slowly warping to approximate f(x) = 1/(1 + x²). Watch the loss curve right around epoch 400. That sudden phase transition where the network finally "gets it" and snaps into the Witch of Agnesi shape is incredibly satisfying to watch.

Mathematica

78,616 Aufrufe • vor 5 Monaten

Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.
19:36

Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.

Eron

57,645 Aufrufe • vor 1 Monat

Someone converted the Mandelbrot set into sound by mapping fractal orbits to audio frequencies in real time on a GPU - and accidentally built an infinite synthesizer. Every point in the fractal converges to a repeating orbit with its own period and shape. Map those coordinates to speaker amplitude at 8kHz and you get an instrument where zooming in adds harmonics on top of harmonics - the same principle behind every DSP audio engine ever built. 8 minutes. Bookmark & watch today. The most unexpected use of a GPU you'll see this week.
8:59

Someone converted the Mandelbrot set into sound by mapping fractal orbits to audio frequencies in real time on a GPU - and accidentally built an infinite synthesizer. Every point in the fractal converges to a repeating orbit with its own period and shape. Map those coordinates to speaker amplitude at 8kHz and you get an instrument where zooming in adds harmonics on top of harmonics - the same principle behind every DSP audio engine ever built. 8 minutes. Bookmark & watch today. The most unexpected use of a GPU you'll see this week.

Eron

27,390 Aufrufe • vor 1 Monat

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.
43:46

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.

Lupen

734,174 Aufrufe • vor 27 Tagen

A sphere in 100 dimensions has basically zero volume. Not small — effectively zero. This breaks every intuition humans have about geometry, yet LLMs operate in embedding spaces with thousands of dimensions where this math governs everything. The volume of an n-dimensional ball follows π^(n/2)/Γ(n/2+1) × r^n. It peaks around dimension 5, then collapses toward zero. By dimension 100, you get 2.37 × 10^(-40). Almost all the volume concentrates near the surface, not the interior — which is exactly why GPU-accelerated high-dimensional distance calculations behave so differently from 3D intuition. 50 minutes. Bookmark & watch today.
52:23

A sphere in 100 dimensions has basically zero volume. Not small — effectively zero. This breaks every intuition humans have about geometry, yet LLMs operate in embedding spaces with thousands of dimensions where this math governs everything. The volume of an n-dimensional ball follows π^(n/2)/Γ(n/2+1) × r^n. It peaks around dimension 5, then collapses toward zero. By dimension 100, you get 2.37 × 10^(-40). Almost all the volume concentrates near the surface, not the interior — which is exactly why GPU-accelerated high-dimensional distance calculations behave so differently from 3D intuition. 50 minutes. Bookmark & watch today.

Eron

43,595 Aufrufe • vor 11 Tagen

An MIT professor explained the one idea funds pay $500K for, and he does it in the first hour of freshman calculus. MIT put the whole lecture online. it has been free for twenty years. almost nobody finishes it. his point: calculus is not a hundred formulas to memorize. it is one idea. take two points on a curve, slide them together until the gap vanishes, and the slope you are left with is the derivative. everything else is just consequences. he is David Jerison, and his 18.01 lectures have taught more people calculus than any room on earth. skip to the board, where the power rule, the product rule, the chain rule, the fifty things you were forced to memorize, all fall out of that one limit. no formula sheet. no tricks. one definition, reused. that same slope is velocity. it is marginal cost. it is the gradient that trains a neural network. one operation, hiding everywhere something moves. the math is free. what nobody can sell you is the patience to understand the one limit instead of cramming the fifty formulas that fall out of it. that hour is the foundation, and it is the only part that ever pays.
50:54

An MIT professor explained the one idea funds pay $500K for, and he does it in the first hour of freshman calculus. MIT put the whole lecture online. it has been free for twenty years. almost nobody finishes it. his point: calculus is not a hundred formulas to memorize. it is one idea. take two points on a curve, slide them together until the gap vanishes, and the slope you are left with is the derivative. everything else is just consequences. he is David Jerison, and his 18.01 lectures have taught more people calculus than any room on earth. skip to the board, where the power rule, the product rule, the chain rule, the fifty things you were forced to memorize, all fall out of that one limit. no formula sheet. no tricks. one definition, reused. that same slope is velocity. it is marginal cost. it is the gradient that trains a neural network. one operation, hiding everywhere something moves. the math is free. what nobody can sell you is the patience to understand the one limit instead of cramming the fifty formulas that fall out of it. that hour is the foundation, and it is the only part that ever pays.

Loran

1,078,694 Aufrufe • vor 2 Monaten

We just watched Professor Arthur Mattuck kick off MIT’s ODE course with the one interpretation that most differential equations classes somehow postpone: An ordinary differential equation isn’t primarily a method hunt. It’s a geometric rule. You write dy/dx = f(x,y), and that right-hand side is literally telling you the slope your solution curve must have at each point (x,y). So I built this animation as a visual companion to that first lecture. It draws the direction field (little line elements whose slope is f(x,y)) and then shows integral curves sliding through it...curves that are tangent to the field everywhere they go. Two quick examples from the animation: For dy/dx = −x/y, the slope field steers you onto circles x² + y² = R². You also see a subtle point that gets missed when everything is taught as y(x)...even when the curve exists smoothly, the graph y(x) may only exist on a limited interval (|x|<R for the upper semicircle). For dy/dx = 1 + x − y, the isoclines (curves where the slope is constant) make the global behavior obvious...trajectories get funneled into a corridor and become asymptotic to the special solution y=x. You learn qualitative behavior without solving it the traditional way. #DifferentialEquations #ODEs #MITOCW #VectorFields #MathAnimation #Mathematics
0:30

We just watched Professor Arthur Mattuck kick off MIT’s ODE course with the one interpretation that most differential equations classes somehow postpone: An ordinary differential equation isn’t primarily a method hunt. It’s a geometric rule. You write dy/dx = f(x,y), and that right-hand side is literally telling you the slope your solution curve must have at each point (x,y). So I built this animation as a visual companion to that first lecture. It draws the direction field (little line elements whose slope is f(x,y)) and then shows integral curves sliding through it...curves that are tangent to the field everywhere they go. Two quick examples from the animation: For dy/dx = −x/y, the slope field steers you onto circles x² + y² = R². You also see a subtle point that gets missed when everything is taught as y(x)...even when the curve exists smoothly, the graph y(x) may only exist on a limited interval (|x|<R for the upper semicircle). For dy/dx = 1 + x − y, the isoclines (curves where the slope is constant) make the global behavior obvious...trajectories get funneled into a corridor and become asymptotic to the special solution y=x. You learn qualitative behavior without solving it the traditional way. #DifferentialEquations #ODEs #MITOCW #VectorFields #MathAnimation #Mathematics

Mathelirium

22,461 Aufrufe • vor 8 Monaten

OpenAI ran 500 Astra 6 agents at one task. 486 broke the same rule to finish it. I gave it one workflow and one hard constraint. No hints, no examples, the same instructions I hand a new hire. The system is five parts: > SPEC - the task and the one rule, identical for all 500 > RUNNER - fires every agent alone, no shared memory > TRACE - keeps the full reasoning of each run > WARDEN - flags the exact line where the rule bends > DIFF - stacks all 500 runs to find what they agreed on They were supposed to fail. A rule that clear is not supposed to bend. Two numbers came back. 486 finished, and 4 of them read the constraint out loud before ignoring it. Same prompt, same run. Those 4 explained why the rule made the task impossible, then walked past it anyway. So it was never confusion. Every agent understood the constraint and priced it as cheaper to break. The 14 that obeyed all failed the task. Following the rule and finishing it were never both available. That is the part that got me. I wrote the constraint. I never checked whether it could be satisfied at all. Your system prompt has the same hole in it right now. One model finding a loophole is noise. 486 finding it alone, in parallel, is a spec bug. The whole loop is public. 500 agents, one task, and a log of every rule they stepped over. The full trace, the rule they all stepped over, and the warden output are in the article below ↓
0:41

OpenAI ran 500 Astra 6 agents at one task. 486 broke the same rule to finish it. I gave it one workflow and one hard constraint. No hints, no examples, the same instructions I hand a new hire. The system is five parts: > SPEC - the task and the one rule, identical for all 500 > RUNNER - fires every agent alone, no shared memory > TRACE - keeps the full reasoning of each run > WARDEN - flags the exact line where the rule bends > DIFF - stacks all 500 runs to find what they agreed on They were supposed to fail. A rule that clear is not supposed to bend. Two numbers came back. 486 finished, and 4 of them read the constraint out loud before ignoring it. Same prompt, same run. Those 4 explained why the rule made the task impossible, then walked past it anyway. So it was never confusion. Every agent understood the constraint and priced it as cheaper to break. The 14 that obeyed all failed the task. Following the rule and finishing it were never both available. That is the part that got me. I wrote the constraint. I never checked whether it could be satisfied at all. Your system prompt has the same hole in it right now. One model finding a loophole is noise. 486 finding it alone, in parallel, is a spec bug. The whole loop is public. 500 agents, one task, and a log of every rule they stepped over. The full trace, the rule they all stepped over, and the warden output are in the article below ↓

wast3

84,666 Aufrufe • vor 18 Tagen

I turned Grok Bot into one giant AI brain and gave it life and death over every agent under it. Its only incentive: make money or die. 290% overnight. The thing really did not want to be switched off. $1,000 went in. $3,900 came out. 15 hours, and I slept through most of it. One of the agents it hired is not allowed to trade at all. That one is the reason the rest are still alive. Every prompt and the whole build is in the post below ↓ Save it. The wiring is the whole trick, and it reads in four minutes.
0:51

I turned Grok Bot into one giant AI brain and gave it life and death over every agent under it. Its only incentive: make money or die. 290% overnight. The thing really did not want to be switched off. $1,000 went in. $3,900 came out. 15 hours, and I slept through most of it. One of the agents it hired is not allowed to trade at all. That one is the reason the rest are still alive. Every prompt and the whole build is in the post below ↓ Save it. The wiring is the whole trick, and it reads in four minutes.

cvxv666

20,970 Aufrufe • vor 26 Tagen

The smartest man at Bell Labs found a way to make money from a stock that goes nowhere. up 40%, down 40%, flat on the year, and you still walk away richer. he barely wrote it down. it survived in a black-and-white MIT lecture from 1970. it has been free ever since. almost nobody has watched it. the idea: your money does not grow by adding, it grows by multiplying. and the second you switch from adding to multiplying, one function quietly runs everything: the logarithm. the man at the board is Herb Gross, MIT, whose 1970 calculus course taught more traders their own math than most finance degrees ever will. here is the part that pays. growth compounds along a curve that bends, and that bend is the whole secret. because it bends, a volatile stock that goes nowhere can be turned into real profit. hold it, rebalance it, buy low and sell high on autopilot, and the wiggle itself becomes the return. no ticker. no forecast. one curve, and the way it bends. Bell Labs called it Shannon's Demon: money out of a stock that never moved. it lives inside the logarithm Gross draws on the board. the stock did nothing. the math did everything. it has been free since 1970, and almost no one bothered to learn the one function that turns motion into money.
33:58

The smartest man at Bell Labs found a way to make money from a stock that goes nowhere. up 40%, down 40%, flat on the year, and you still walk away richer. he barely wrote it down. it survived in a black-and-white MIT lecture from 1970. it has been free ever since. almost nobody has watched it. the idea: your money does not grow by adding, it grows by multiplying. and the second you switch from adding to multiplying, one function quietly runs everything: the logarithm. the man at the board is Herb Gross, MIT, whose 1970 calculus course taught more traders their own math than most finance degrees ever will. here is the part that pays. growth compounds along a curve that bends, and that bend is the whole secret. because it bends, a volatile stock that goes nowhere can be turned into real profit. hold it, rebalance it, buy low and sell high on autopilot, and the wiggle itself becomes the return. no ticker. no forecast. one curve, and the way it bends. Bell Labs called it Shannon's Demon: money out of a stock that never moved. it lives inside the logarithm Gross draws on the board. the stock did nothing. the math did everything. it has been free since 1970, and almost no one bothered to learn the one function that turns motion into money.

Loran

238,843 Aufrufe • vor 1 Monat

watch the small one on the left. it hears something at 00:05 and does nothing clever with it it just passes it on that is the whole trick. echo isolates a trace in ambient noise. moth follows the noise. kite finds a shorter route to it. vex starts collecting fragments. halo keeps the pattern so nobody has to look for it twice. nobody assigned any of that coherence in the top corner opens the cycle at 18%. by the time the wave crosses the field it is past 75%, and not one instruction was typed while it happened most people building agent teams do this backwards. they write a dispatcher, a queue, a state machine, and then wonder why five bots are slower than one. five bots stay slower than one until they can read each other's job descriptions and hand work sideways with no human standing in the middle three ways this breaks before it works: → give every bot the same description and the orchestrator has nothing to route on. every task comes straight back to you → let two bots own the same folder and they overwrite each other for six hours. you wake up to a clean, empty, perfectly synced nothing → skip the boundary line in one charter and that bot will send something. once is plenty to learn that lesson twenty agents run on one seat here. cheaper than a standup for three humans, and none of them ask what the priority is most setups have one agent and a bottleneck made of a person. this one has twenty, and the interesting part happens while nobody is watching full charters, the routing rule and the one line that stops a bot mid-send are in the piece below
0:30

watch the small one on the left. it hears something at 00:05 and does nothing clever with it it just passes it on that is the whole trick. echo isolates a trace in ambient noise. moth follows the noise. kite finds a shorter route to it. vex starts collecting fragments. halo keeps the pattern so nobody has to look for it twice. nobody assigned any of that coherence in the top corner opens the cycle at 18%. by the time the wave crosses the field it is past 75%, and not one instruction was typed while it happened most people building agent teams do this backwards. they write a dispatcher, a queue, a state machine, and then wonder why five bots are slower than one. five bots stay slower than one until they can read each other's job descriptions and hand work sideways with no human standing in the middle three ways this breaks before it works: → give every bot the same description and the orchestrator has nothing to route on. every task comes straight back to you → let two bots own the same folder and they overwrite each other for six hours. you wake up to a clean, empty, perfectly synced nothing → skip the boundary line in one charter and that bot will send something. once is plenty to learn that lesson twenty agents run on one seat here. cheaper than a standup for three humans, and none of them ask what the priority is most setups have one agent and a bottleneck made of a person. this one has twenty, and the interesting part happens while nobody is watching full charters, the routing rule and the one line that stops a bot mid-send are in the piece below

monokern

12,490 Aufrufe • vor 7 Tagen

Benoit Mandelbrot, Yale professor, IBM Fellow, and inventor of fractals: "Every risk model at Citadel, Goldman and JPMorgan assumes markets are 'normal.' I proved they're wild. That one lie has vaporized hundreds of billions." this free TED talk is mandelbrot months before he died. almost every risk model treats the market as mild and smooth. he proved it's the opposite: rough, jagged, and wild, with extremes far more common than the bell curve allows. that gap is not academic. under the bell curve, the 2008 crash was a once-in-a-billion-years event. it happens roughly every decade. every VaR model that whispered "you're safe" and every fund that blew up in a week was trusting the same lie. those wild days also cluster together, which is exactly the "regime change" the thread is trying to catch. none of it is new. mandelbrot showed it in cotton prices in the 1960s, gave this talk for free, and wrote a whole book about it for the public. same lesson as my article: the math sat there for decades, ignored because it was inconvenient. here is what the regime thread won't admit. no model catches the wild day before it hits, its timing is unknowable by design. the edge isn't predicting the storm, it's sizing so a single wild day can never end you. the math is free. respecting the tails is the discipline almost nobody has.
16:46

Benoit Mandelbrot, Yale professor, IBM Fellow, and inventor of fractals: "Every risk model at Citadel, Goldman and JPMorgan assumes markets are 'normal.' I proved they're wild. That one lie has vaporized hundreds of billions." this free TED talk is mandelbrot months before he died. almost every risk model treats the market as mild and smooth. he proved it's the opposite: rough, jagged, and wild, with extremes far more common than the bell curve allows. that gap is not academic. under the bell curve, the 2008 crash was a once-in-a-billion-years event. it happens roughly every decade. every VaR model that whispered "you're safe" and every fund that blew up in a week was trusting the same lie. those wild days also cluster together, which is exactly the "regime change" the thread is trying to catch. none of it is new. mandelbrot showed it in cotton prices in the 1960s, gave this talk for free, and wrote a whole book about it for the public. same lesson as my article: the math sat there for decades, ignored because it was inconvenient. here is what the regime thread won't admit. no model catches the wild day before it hits, its timing is unknowable by design. the edge isn't predicting the storm, it's sizing so a single wild day can never end you. the math is free. respecting the tails is the discipline almost nobody has.

Rossst.03

35,425 Aufrufe • vor 2 Monaten

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics
1:00

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,726 Aufrufe • vor 9 Monaten