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The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea...

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Mathelirium

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31,688 просмотров • 7 месяцев назад •via X (Twitter)

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Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation
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Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation

Mathelirium

100,544 просмотров • 9 месяцев назад

Brownian Motion Has a Hidden Speed Limit Once you’ve come to terms with Brownian motion having no derivative, acquiring a deterministic Quadratic Variation, and crossing zero infinitely often in every neighbourhood, the next surprise is the Law of The Iterated Logarithm. A Brownian path may look completely unconstrained, but its largest excursions obey an astonishingly sharp envelope. The right scale is not merely √t, but √[2t log log t], and that strange log log t term is exactly what sets the true boundary for how far Bₜ can wander. The theorem says that, almost surely, the path comes arbitrarily close to the upper and lower curves ±√[2t log log t] infinitely many times, yet does not genuinely live beyond them. Choose a smaller multiple and Bₜ will break through your barrier infinitely often. Choose a larger one and, beyond some point, it almost never reaches it again. #BrownianMotion #LawOfTheIteratedLogarithm #StochasticProcesses #RandomPaths #ProbabilityTheory #FractalBehaviour
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Brownian Motion Has a Hidden Speed Limit Once you’ve come to terms with Brownian motion having no derivative, acquiring a deterministic Quadratic Variation, and crossing zero infinitely often in every neighbourhood, the next surprise is the Law of The Iterated Logarithm. A Brownian path may look completely unconstrained, but its largest excursions obey an astonishingly sharp envelope. The right scale is not merely √t, but √[2t log log t], and that strange log log t term is exactly what sets the true boundary for how far Bₜ can wander. The theorem says that, almost surely, the path comes arbitrarily close to the upper and lower curves ±√[2t log log t] infinitely many times, yet does not genuinely live beyond them. Choose a smaller multiple and Bₜ will break through your barrier infinitely often. Choose a larger one and, beyond some point, it almost never reaches it again. #BrownianMotion #LawOfTheIteratedLogarithm #StochasticProcesses #RandomPaths #ProbabilityTheory #FractalBehaviour

Mathelirium

29,105 просмотров • 6 месяцев назад

Today we introduce Stochastic Differential Equations (SDEs). I find that the best way to introduce these complex concepts is to look at an application. This is part I of the lecture🙂 We look at the theory behind electromagnetic scattering/radar clutter which leads to anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum...at time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where the magic of Brownian motion appears naturally and in the most reasonable way. Think of all the microscopic chaos...tiny motions, index fluctuations, path jitters, Doppler shifts that shows up as small random kicks to the phases θⱼ(t) over very short times. If you just made θⱼ(t) random in an ad-hoc way (say, resampling independent angles at each time), the field would jump around unrealistically with no temporal structure. Brownian motion is what you get when you let each phase take the continuous-time limit of many tiny, independent kicks...it’s continuous in t, it has the right cumulative variance growth, and it remembers just enough of its past to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing...it’s the clean way to compress all the small random stuff into a single process that actually matches how the phases wander in time. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing
0:30

Today we introduce Stochastic Differential Equations (SDEs). I find that the best way to introduce these complex concepts is to look at an application. This is part I of the lecture🙂 We look at the theory behind electromagnetic scattering/radar clutter which leads to anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum...at time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where the magic of Brownian motion appears naturally and in the most reasonable way. Think of all the microscopic chaos...tiny motions, index fluctuations, path jitters, Doppler shifts that shows up as small random kicks to the phases θⱼ(t) over very short times. If you just made θⱼ(t) random in an ad-hoc way (say, resampling independent angles at each time), the field would jump around unrealistically with no temporal structure. Brownian motion is what you get when you let each phase take the continuous-time limit of many tiny, independent kicks...it’s continuous in t, it has the right cumulative variance growth, and it remembers just enough of its past to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing...it’s the clean way to compress all the small random stuff into a single process that actually matches how the phases wander in time. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

55,319 просмотров • 9 месяцев назад

Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing
0:30

Today we introduce Stochastic Differential Equations (SDEs), and the main thing to watch for is this: We’ll use Brownian motion as the basic noise source, then see how well-known SDEs drop out of it naturally, without guessing. I still think the best way into these concepts is through an application. We look at the theory behind electromagnetic scattering and radar clutter, which leads straight into anomaly detection on scattering statistics. When a narrowband wave scatters off a messy cloud of particles, the complex field at your receiver is a random phasor sum. At time t you can write the electric field as E_N(t) = Σⱼ₌₁ᴺ e^{iθⱼ(t)}, each term a unit arrow in the complex plane from scatterer j. This is exactly where Brownian motion shows up in the most reasonable way. Think of all the microscopic chaos: tiny motions, index fluctuations, path jitters, Doppler shifts. Over short times, all of that shows up as small random kicks to the phases θⱼ(t). If you made θⱼ(t) random in an ad-hoc way, like resampling a fresh independent angle at every instant, the field would jump around unrealistically with no physical time structure. Brownian motion is what you get when each phase takes the continuous-time limit of many tiny, independent kicks. It’s continuous in t, its variance grows the right way, and it carries just enough temporal structure to look physical. So we model each phase as a Brownian walk, θⱼ(t) = θⱼ⁰ + σ_θ Bⱼ(t), with independent Brownian motions Bⱼ(t) and a phase-diffusion rate σ_θ. Brownian motion here isn’t window dressing. It’s the clean way to compress all the small random stuff into a single process that actually matches how phases wander in time. This is called Rayleigh Scattering, but the same sum of many tiny coherent echoes shows up in lots of places...e.g. wireless multipath fading (phones/Wi-Fi), laser/optical links through atmospheric turbulence, ultrasound speckle in tissue, and sonar/underwater acoustics in rough or bubbly water. #StochasticProcesses #BrownianMotion #ItoCalculus #RadarClutter #RayleighScattering #SignalProcessing

Mathelirium

31,182 просмотров • 7 месяцев назад

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
0:30

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 просмотров • 3 месяцев назад

A Japanese mathematician published a result in 1944 that nobody understood for twenty years. Today it runs inside every options desk on Wall Street. Goldman pays $400K to quants who can derive it from scratch and explain why classical calculus gives the wrong answer without it. His name is Choongbum Lee. MIT, 18.S096, Topics in Mathematics with Applications in Finance. The course that Wall Street watches. This is lecture 17. It derives Ito's Lemma from scratch. He opens with the problem nobody in classical calculus can solve. Then the foundation. Brownian motion is the limit of a random walk taken to infinity. Each trade pushes a price up or down by a tiny amount. A million trades a day. The limit of that process is Brownian motion. Einstein proved this for pollen particles in 1905. The finance world borrowed the math fifty years later. Then three properties that make no sense until you see them derived. Brownian motion crosses zero infinitely often. It never escapes to infinity. And it is nowhere differentiable - with probability one, every path is continuous but has no slope at any point. That last property is why classical calculus breaks completely. Then quadratic variation. For any smooth function, chop an interval into n pieces, square the increments, sum them - the result goes to zero. For Brownian motion it goes to T. The increments are too wild to vanish. That single fact is why Ito's Lemma has a second term that classical calculus does not. Watch the moment he derives it. Taylor expansion applied to a function of Brownian motion. The first term is what you expect. The second term appears precisely because the squared increment does not vanish. Without it, options pricing gives wrong answers. With it, you have Black-Scholes. A quant I know sends this lecture to junior analysts who cannot explain why their pricing model drifts. Says it fixes in ninety minutes what two years of finance courses left open. Free on YouTube, MIT OpenCourseWare, 18.S096. bookmark this and watch later - the math behind every options desk on Wall Street fits on one blackboard, and this is the lecture that shows you why
1:12:31

A Japanese mathematician published a result in 1944 that nobody understood for twenty years. Today it runs inside every options desk on Wall Street. Goldman pays $400K to quants who can derive it from scratch and explain why classical calculus gives the wrong answer without it. His name is Choongbum Lee. MIT, 18.S096, Topics in Mathematics with Applications in Finance. The course that Wall Street watches. This is lecture 17. It derives Ito's Lemma from scratch. He opens with the problem nobody in classical calculus can solve. Then the foundation. Brownian motion is the limit of a random walk taken to infinity. Each trade pushes a price up or down by a tiny amount. A million trades a day. The limit of that process is Brownian motion. Einstein proved this for pollen particles in 1905. The finance world borrowed the math fifty years later. Then three properties that make no sense until you see them derived. Brownian motion crosses zero infinitely often. It never escapes to infinity. And it is nowhere differentiable - with probability one, every path is continuous but has no slope at any point. That last property is why classical calculus breaks completely. Then quadratic variation. For any smooth function, chop an interval into n pieces, square the increments, sum them - the result goes to zero. For Brownian motion it goes to T. The increments are too wild to vanish. That single fact is why Ito's Lemma has a second term that classical calculus does not. Watch the moment he derives it. Taylor expansion applied to a function of Brownian motion. The first term is what you expect. The second term appears precisely because the squared increment does not vanish. Without it, options pricing gives wrong answers. With it, you have Black-Scholes. A quant I know sends this lecture to junior analysts who cannot explain why their pricing model drifts. Says it fixes in ninety minutes what two years of finance courses left open. Free on YouTube, MIT OpenCourseWare, 18.S096. bookmark this and watch later - the math behind every options desk on Wall Street fits on one blackboard, and this is the lecture that shows you why

Lupen

66,176 просмотров • 26 дней назад

If you can get to the path of totality for the #solareclipse2024 you should. It is literally a night and day difference from 99% to 100% coverage. It is a life-changing experience. Here's a look at the path of 100% totality through the middle of the U.S.
0:24

If you can get to the path of totality for the #solareclipse2024 you should. It is literally a night and day difference from 99% to 100% coverage. It is a life-changing experience. Here's a look at the path of 100% totality through the middle of the U.S.

Brad Panovich

1,526,539 просмотров • 2 лет назад

What Curve Beats a Straight Line? Take two fixed points A and B. A bead slides without friction under gravity. Among all possible curves connecting A to B, which one gives the smallest travel time? This is the Brachistochrone problem. The time functional is T[y] = ∫ √((1 + y′²)/(2gy)) dx where y is the vertical drop from the starting point. The winning curve is not a straight line. It is a cycloid x(θ) = a(θ - sin θ) y(θ) = a(1 - cos θ) The bead first falls steeply to gain speed, then uses that speed across the remaining distance. The path is longer than the straight segment, yet the travel time is smaller. The shortest path solves geometry. The fastest path solves variation. #CalculusOfVariations #Brachistochrone #Cycloid #ClassicalMechanics #MathematicalPhysics #Optimization #DifferentialEquations #PhysicsAnimation #Mathematics
0:55

What Curve Beats a Straight Line? Take two fixed points A and B. A bead slides without friction under gravity. Among all possible curves connecting A to B, which one gives the smallest travel time? This is the Brachistochrone problem. The time functional is T[y] = ∫ √((1 + y′²)/(2gy)) dx where y is the vertical drop from the starting point. The winning curve is not a straight line. It is a cycloid x(θ) = a(θ - sin θ) y(θ) = a(1 - cos θ) The bead first falls steeply to gain speed, then uses that speed across the remaining distance. The path is longer than the straight segment, yet the travel time is smaller. The shortest path solves geometry. The fastest path solves variation. #CalculusOfVariations #Brachistochrone #Cycloid #ClassicalMechanics #MathematicalPhysics #Optimization #DifferentialEquations #PhysicsAnimation #Mathematics

Mathelirium

17,600 просмотров • 2 месяцев назад

Viktor Hovland made 8 birdies to close with a 5 under par 67 at the Masters. After the round, he went deep on the ball flight laws and the importance of matching up the club face to the swing path: “It's a good question. It's just tough to answer. First of all, that's why I like TrackMan and understanding the ball flight laws are very good to have because you can start a ball left, more left than you would like, but if it's curving back, who cares. That just means you're -- granted, if you're hitting it in the center your club path is just maybe creeping a bit more left, but if your face is in the right spot you're just hitting a bigger curve and then you can adjust that accordingly. “If you're starting the ball left and it's going left, is it a face issue, are you striking it off the heel and they're not coming back? There's a lot of minutiae involved with that. “If it's just a face-to-path issue or a club path direction, you can solve that pretty easily by playing a bigger cut or playing a bigger draw or moving it a bit more forward or back in the stance, simple stuff like that.” This is a great explanation that a lot of amateurs can learn from. The Masters
1:01

Viktor Hovland made 8 birdies to close with a 5 under par 67 at the Masters. After the round, he went deep on the ball flight laws and the importance of matching up the club face to the swing path: “It's a good question. It's just tough to answer. First of all, that's why I like TrackMan and understanding the ball flight laws are very good to have because you can start a ball left, more left than you would like, but if it's curving back, who cares. That just means you're -- granted, if you're hitting it in the center your club path is just maybe creeping a bit more left, but if your face is in the right spot you're just hitting a bigger curve and then you can adjust that accordingly. “If you're starting the ball left and it's going left, is it a face issue, are you striking it off the heel and they're not coming back? There's a lot of minutiae involved with that. “If it's just a face-to-path issue or a club path direction, you can solve that pretty easily by playing a bigger cut or playing a bigger draw or moving it a bit more forward or back in the stance, simple stuff like that.” This is a great explanation that a lot of amateurs can learn from. The Masters

Flushing It

667,750 просмотров • 5 месяцев назад

Have you ever looked closely at the Amazon River? 🤔 One of the longest river systems on Earth… flowing in a single, consistent direction for over 43 hundred miles. If Earth is a spinning globe, How does water maintain a continuous level path Across curvature, rotation, and changing orientations? Water always seeks equilibrium. So what exactly is it responding to?
1:35

Have you ever looked closely at the Amazon River? 🤔 One of the longest river systems on Earth… flowing in a single, consistent direction for over 43 hundred miles. If Earth is a spinning globe, How does water maintain a continuous level path Across curvature, rotation, and changing orientations? Water always seeks equilibrium. So what exactly is it responding to?

vegastar

80,373 просмотров • 9 месяцев назад

Today I stand in the city where my grandfather Shaheed Zulfikar Ali Bhutto was judicially murdered. At the very park my mother, Shaheed Mohtarma Benazir Bhutto, addressed her final jalsa. I stand here because Pakistan is on the brink of disaster once again. There are two paths before you, one is the path our country has walked for decades. It is a path that leads to hatred and division. The other is a new path, a path that puts you, the people, at the center of politics. It is the path that will lead us all to a better tomorrow. On 8th February vote for the arrow of Shaheed Mohtarma Benazir Bhutto, and choose a new path to peace, progress, and prosperity #چنو_نئی_سوچ_کو
0:42

Today I stand in the city where my grandfather Shaheed Zulfikar Ali Bhutto was judicially murdered. At the very park my mother, Shaheed Mohtarma Benazir Bhutto, addressed her final jalsa. I stand here because Pakistan is on the brink of disaster once again. There are two paths before you, one is the path our country has walked for decades. It is a path that leads to hatred and division. The other is a new path, a path that puts you, the people, at the center of politics. It is the path that will lead us all to a better tomorrow. On 8th February vote for the arrow of Shaheed Mohtarma Benazir Bhutto, and choose a new path to peace, progress, and prosperity #چنو_نئی_سوچ_کو

Bilawal Bhutto Zardari

267,558 просмотров • 2 лет назад

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 просмотров • 8 месяцев назад

NEW: 🇺🇸 "This is money... Of course, it needs to have a regulated path," says BitGo CEO Mike Belshe. Belshe warns that without a framework, the U.S. cannot begin the process: "If you don't put the CLARITY Act in or something like it, then you don't even get to get started." 📜
0:53

NEW: 🇺🇸 "This is money... Of course, it needs to have a regulated path," says BitGo CEO Mike Belshe. Belshe warns that without a framework, the U.S. cannot begin the process: "If you don't put the CLARITY Act in or something like it, then you don't even get to get started." 📜

Bitcoin.com News

70,677 просмотров • 26 дней назад

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 просмотров • 9 месяцев назад

Dear Community, I hear you. But I am not just parroting Trump. I ask you to think through my logic. Think from first principles. Poke holes in the logic where you see them. Let's have an honest back and forth over the issues, and either find our way logically to the path this country is already on - or discover a better path and shout it out. Make no mistake, I know that this is just a game of chicken. And chicken, is chicken, because it puts us on a collision course head-on. We will see if it works. But anybody -> what's a better path? Share it with us.
1:44

Dear Community, I hear you. But I am not just parroting Trump. I ask you to think through my logic. Think from first principles. Poke holes in the logic where you see them. Let's have an honest back and forth over the issues, and either find our way logically to the path this country is already on - or discover a better path and shout it out. Make no mistake, I know that this is just a game of chicken. And chicken, is chicken, because it puts us on a collision course head-on. We will see if it works. But anybody -> what's a better path? Share it with us.

Tom Bilyeu

16,832 просмотров • 1 год назад

Blender Help Needed! Doing this Shortest Edge path setup, but when animating the trim curve it highlights all these duplicates in the curve, how to i get rid of that. adding a merge by distance ruins the whole effect i am after. which is that line growth effect.
0:23

Blender Help Needed! Doing this Shortest Edge path setup, but when animating the trim curve it highlights all these duplicates in the curve, how to i get rid of that. adding a merge by distance ruins the whole effect i am after. which is that line growth effect.

Ducky 3D

17,515 просмотров • 11 месяцев назад

007/100 Buttons. This button was way more complex than I first thought. I love the page transition on the Truus site that Dennis Snellenberg and Jordan Gilroy worked on. So I wanted to use that drawing SVG Path idea somehow and fit it into a button. In best case with a mask effect. The idea was to have an SVG path in the background that fills the button. At the same time, this path should also be used as a mask to reveal the differently colored text. The SVG should also be replaceable and work with other SVGs. For the path animation, I can use the DrawSVGPlugin from GSAP. Sounds like a solid plan. I built the button so far, and it worked great in Chrome and Firefox. In Safari, nope. I used an SVG mask that I referenced through CSS. But Safari can’t handle that properly when the path inside the mask is animated, so the animation lags. There was no simple solution for that. At least I didn’t find something. So I had to rebuild the button in another way. What I found was that the SVG mask animation works in Safari when the mask is placed on the desired element inside the SVG. That’s where foreignObject comes in, it allows you to use normal HTML elements inside an SVG. Using that, I rebuilt the hover text inside the SVG. I then referenced the mask on the foreignObject with a bit of JavaScript. And it worked! The button’s structure looks more complicated than I wanted it to be, but that happens quite often when you need to make something work across different browsers 😃 Crafting 100 Buttons with Osmo ⏳ Total time: 96h
0:35

007/100 Buttons. This button was way more complex than I first thought. I love the page transition on the Truus site that Dennis Snellenberg and Jordan Gilroy worked on. So I wanted to use that drawing SVG Path idea somehow and fit it into a button. In best case with a mask effect. The idea was to have an SVG path in the background that fills the button. At the same time, this path should also be used as a mask to reveal the differently colored text. The SVG should also be replaceable and work with other SVGs. For the path animation, I can use the DrawSVGPlugin from GSAP. Sounds like a solid plan. I built the button so far, and it worked great in Chrome and Firefox. In Safari, nope. I used an SVG mask that I referenced through CSS. But Safari can’t handle that properly when the path inside the mask is animated, so the animation lags. There was no simple solution for that. At least I didn’t find something. So I had to rebuild the button in another way. What I found was that the SVG mask animation works in Safari when the mask is placed on the desired element inside the SVG. That’s where foreignObject comes in, it allows you to use normal HTML elements inside an SVG. Using that, I rebuilt the hover text inside the SVG. I then referenced the mask on the foreignObject with a bit of JavaScript. And it worked! The button’s structure looks more complicated than I wanted it to be, but that happens quite often when you need to make something work across different browsers 😃 Crafting 100 Buttons with Osmo ⏳ Total time: 96h

Eduard Bodak

67,623 просмотров • 4 месяцев назад

🚨🇮🇷🇺🇸IRAN TO U.S: BOMB US, AND WE GO NUCLEAR Adviser to Iran’s Supreme Leader, Ali Larijani: “We have said that there is a Fatwa, we are working under the supervision of the IAEA and we won’t move towards a nuclear weapon, but if you make a mistake regarding Iran’s nuclear issue, you’ll force Iran to take that path, because it has to defend itself. If you at some point take the path of bombing (Iran’s nuclear sites) yourself or through Israel, you’ll force Iran to make a different decision. This is not your betterment.” Source: AP News
0:26

🚨🇮🇷🇺🇸IRAN TO U.S: BOMB US, AND WE GO NUCLEAR Adviser to Iran’s Supreme Leader, Ali Larijani: “We have said that there is a Fatwa, we are working under the supervision of the IAEA and we won’t move towards a nuclear weapon, but if you make a mistake regarding Iran’s nuclear issue, you’ll force Iran to take that path, because it has to defend itself. If you at some point take the path of bombing (Iran’s nuclear sites) yourself or through Israel, you’ll force Iran to make a different decision. This is not your betterment.” Source: AP News

Mario Nawfal

493,034 просмотров • 1 год назад

If you had to choose one, nutrition or movement, what would it be? For years, we’ve treated it like the chicken-or-egg dilemma. But if we zoom out and look at physiology, the answer starts pointing in one direction. All roads may lead to Rome, but metabolic efficiency, sustainability and long-term health follow a clearer path. A conversation with Alan Couzens Aitor Viribay Links below 🧵 👇
0:59

If you had to choose one, nutrition or movement, what would it be? For years, we’ve treated it like the chicken-or-egg dilemma. But if we zoom out and look at physiology, the answer starts pointing in one direction. All roads may lead to Rome, but metabolic efficiency, sustainability and long-term health follow a clearer path. A conversation with Alan Couzens Aitor Viribay Links below 🧵 👇

Iñaki de la Parra

26,137 просмотров • 6 месяцев назад