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There’s a random curve that is perfectly continuous in time, never jumps, models smooth noise in physics and finance… and yet with probability 1 it has no derivative at any point. If you zoom in anywhere, it never straightens out. Brownian motion started as a physics headache long before...

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Mathelirium

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33,072 次观看 • 8 个月前 •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
0:30

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 次观看 • 8 个月前

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation
0:55

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation

Mathelirium

18,616 次观看 • 1 个月前

Have You Ever Heard of Molecular Communication Systems? Instead of using radio waves or light, molecular communications send information carefully through released chemical particles. A transmitter emits molecules into a medium, the channel reshapes that signal through diffusion, flow, and randomness. The receiver reads the message from the molecules that arrive and bind. It is communication theory at microscopic scale, where physics, probability and biology all become part of the channel. Some applications you may know: Targeted drug delivery inside the body, microscopic health monitoring with nanosensors, and lab-on-a-chip.
0:48

Have You Ever Heard of Molecular Communication Systems? Instead of using radio waves or light, molecular communications send information carefully through released chemical particles. A transmitter emits molecules into a medium, the channel reshapes that signal through diffusion, flow, and randomness. The receiver reads the message from the molecules that arrive and bind. It is communication theory at microscopic scale, where physics, probability and biology all become part of the channel. Some applications you may know: Targeted drug delivery inside the body, microscopic health monitoring with nanosensors, and lab-on-a-chip.

Mathelirium

19,452 次观看 • 5 个月前

Russians dominate 20th-century probability theory. It’s not close!🤯 In 1960 the Soviets drop another hammer: Girsanov asks a baffling question: can you steer randomness without touching the randomness? Girsanov’s answer is to stop thinking in forces and start thinking in judgement. You don’t alter the Brownian wiggles...you change which full histories are treated as plausible. Paths that already happen to lean the right way get extra weight. Paths that fight the desired trend get taxed. After that quiet re-scoring, the same swarm suddenly behaves as if a drift has been switched on. This render turns that invisible accounting into cinema. Probability doesn’t move by being pushed, it moves by being selected. The noise is identical; only the definition of typical was rewritten. That trick is everywhere: it’s the backbone of risk-neutral pricing in mathematical finance, a workhorse in nonlinear filtering (changing measures to make observations behave nicely), and a practical engine behind rare-event simulation / importance sampling in reliability and queuing networks. #StochasticProcesses #Girsanov #ProbabilityTheory #MonteCarlo #RareEvents #ProbabilityTheory
0:33

Russians dominate 20th-century probability theory. It’s not close!🤯 In 1960 the Soviets drop another hammer: Girsanov asks a baffling question: can you steer randomness without touching the randomness? Girsanov’s answer is to stop thinking in forces and start thinking in judgement. You don’t alter the Brownian wiggles...you change which full histories are treated as plausible. Paths that already happen to lean the right way get extra weight. Paths that fight the desired trend get taxed. After that quiet re-scoring, the same swarm suddenly behaves as if a drift has been switched on. This render turns that invisible accounting into cinema. Probability doesn’t move by being pushed, it moves by being selected. The noise is identical; only the definition of typical was rewritten. That trick is everywhere: it’s the backbone of risk-neutral pricing in mathematical finance, a workhorse in nonlinear filtering (changing measures to make observations behave nicely), and a practical engine behind rare-event simulation / importance sampling in reliability and queuing networks. #StochasticProcesses #Girsanov #ProbabilityTheory #MonteCarlo #RareEvents #ProbabilityTheory

Mathelirium

76,716 次观看 • 8 个月前

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.
0:33

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 次观看 • 5 个月前

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 次观看 • 8 个月前

Markov Proved that Randomness With Memory Can Still Settle Into Law Markov’s 1906 result was a major turning point in Probability Theory. He showed that even when successive events are dependent, long-run averages can still stabilize. That broke the old habit of tying the Law of Large Numbers only to independence, and opened the door to a much broader Mathematics of Stochastic Systems. From there, the whole subject of Markov Chains grew into one of the central languages of modern probability. It feeds directly into Ergodic Theory, Statistical Physics, Queueing Systems, Population Models, Monte Carlo methods, Hidden Markov models, Speech Recognition, Finance, Genetics, and even web search. The idea is simple and deep at once. It shows that local randomness can still produce global regularity. The paper: А. А. Марков (1906). Распространение закона больших чисел на величины, зависящие друг от друга (Extension of The Law of Large Numbers to Dependent Quantities).
0:29

Markov Proved that Randomness With Memory Can Still Settle Into Law Markov’s 1906 result was a major turning point in Probability Theory. He showed that even when successive events are dependent, long-run averages can still stabilize. That broke the old habit of tying the Law of Large Numbers only to independence, and opened the door to a much broader Mathematics of Stochastic Systems. From there, the whole subject of Markov Chains grew into one of the central languages of modern probability. It feeds directly into Ergodic Theory, Statistical Physics, Queueing Systems, Population Models, Monte Carlo methods, Hidden Markov models, Speech Recognition, Finance, Genetics, and even web search. The idea is simple and deep at once. It shows that local randomness can still produce global regularity. The paper: А. А. Марков (1906). Распространение закона больших чисел на величины, зависящие друг от друга (Extension of The Law of Large Numbers to Dependent Quantities).

Mathelirium

15,338 次观看 • 4 个月前

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 次观看 • 6 个月前

In 1887, Heinrich Hertz noticed that some light made it easier for electrons to escape a metal surface, but the pattern made no sense under Classical Physics. The surprise was that very bright red light could fail, while dim blue or ultraviolet light worked immediately. Einstein explained this in 1905 by proposing that light comes in energy packets, and only packets with enough energy can knock electrons free. That was the Photoelectric Effect, and it helped earn him the 1921 Nobel Prize in Physics.
0:48

In 1887, Heinrich Hertz noticed that some light made it easier for electrons to escape a metal surface, but the pattern made no sense under Classical Physics. The surprise was that very bright red light could fail, while dim blue or ultraviolet light worked immediately. Einstein explained this in 1905 by proposing that light comes in energy packets, and only packets with enough energy can knock electrons free. That was the Photoelectric Effect, and it helped earn him the 1921 Nobel Prize in Physics.

Mathelirium

64,856 次观看 • 5 个月前

🚨 NOW: Stephen Miller just said it PERFECTLY "There has never been in HISTORY a more radical day and night shift in the security posture of a country that would happen to that border between January 20th and January 21st!" "Never in HISTORY has a nation achieved such a complete turnaround in its security posture in so short a time. And that was led and driven out of the Department of War with an urgency and a speed and a ferocity that I've never witnessed before in any other policy area that I had ever worked on." 🇺🇸🇺🇸🇺🇸
1:04

🚨 NOW: Stephen Miller just said it PERFECTLY "There has never been in HISTORY a more radical day and night shift in the security posture of a country that would happen to that border between January 20th and January 21st!" "Never in HISTORY has a nation achieved such a complete turnaround in its security posture in so short a time. And that was led and driven out of the Department of War with an urgency and a speed and a ferocity that I've never witnessed before in any other policy area that I had ever worked on." 🇺🇸🇺🇸🇺🇸

Eric Daugherty

29,798 次观看 • 25 天前

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics
0:56

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics

Mathelirium

91,291 次观看 • 1 个月前

an MIT professor who taught both physics and finance told his class something none of them expected "finance is harder than physics" not as a joke. as a mathematical statement in physics, the laws don't change. gravity works the same today as it did a billion years ago you can run an experiment, get a result, and repeat it forever in finance, the moment you discover a law, the participants learn it too and their behavior changes the system you just measured in physics, electrons don't read your paper and start moving differently in finance, traders do. every published edge gets arbitraged away by the people who read it this is why quant models have a half-life and physics equations don't Newton's laws: 300+ years and counting Long-Term Capital Management's model: worked perfectly until it didn't, lost $4.6 billion in 4 months the system you're modeling is aware of you modeling it that's not a solvable problem. it's a permanent condition and the quants who survive are the ones who build for it instead of pretending it doesn't exist > this lecture: MIT finance series, free, public, 53 seconds > LTCM collapse: 1998, Nobel Prize winners, $4.6B loss > Renaissance's solution: never stop researching, replace signals before they decay > average lifespan of a quant signal: 2-5 years before it's crowded out retail builds one strategy and trades it until it breaks quant desks build a research engine that produces new strategies faster than old ones die that's not a difference in skill. it's a difference in understanding what game you're actually playing full breakdown in the video below
0:53

an MIT professor who taught both physics and finance told his class something none of them expected "finance is harder than physics" not as a joke. as a mathematical statement in physics, the laws don't change. gravity works the same today as it did a billion years ago you can run an experiment, get a result, and repeat it forever in finance, the moment you discover a law, the participants learn it too and their behavior changes the system you just measured in physics, electrons don't read your paper and start moving differently in finance, traders do. every published edge gets arbitraged away by the people who read it this is why quant models have a half-life and physics equations don't Newton's laws: 300+ years and counting Long-Term Capital Management's model: worked perfectly until it didn't, lost $4.6 billion in 4 months the system you're modeling is aware of you modeling it that's not a solvable problem. it's a permanent condition and the quants who survive are the ones who build for it instead of pretending it doesn't exist > this lecture: MIT finance series, free, public, 53 seconds > LTCM collapse: 1998, Nobel Prize winners, $4.6B loss > Renaissance's solution: never stop researching, replace signals before they decay > average lifespan of a quant signal: 2-5 years before it's crowded out retail builds one strategy and trades it until it breaks quant desks build a research engine that produces new strategies faster than old ones die that's not a difference in skill. it's a difference in understanding what game you're actually playing full breakdown in the video below

delost

19,010 次观看 • 2 个月前

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
0:30

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 次观看 • 5 个月前

Russian Genius Found Structure in The Mess of Turbulence In 1941, Andrey Kolmogorov proposed a simple but powerful picture of turbulence: Energy does not disappear where it is injected but moves. Large eddies pass energy to smaller ones, those smaller ones feed even finer motion, and viscosity finally removes it at the tiniest scales. That cascade picture became one of the foundations of modern turbulence theory. The point is not just that turbulence looks messy. The point is that the mess has structure, and Kolmogorov’s great insight was to treat that structure statistically.
0:30

Russian Genius Found Structure in The Mess of Turbulence In 1941, Andrey Kolmogorov proposed a simple but powerful picture of turbulence: Energy does not disappear where it is injected but moves. Large eddies pass energy to smaller ones, those smaller ones feed even finer motion, and viscosity finally removes it at the tiniest scales. That cascade picture became one of the foundations of modern turbulence theory. The point is not just that turbulence looks messy. The point is that the mess has structure, and Kolmogorov’s great insight was to treat that structure statistically.

Mathelirium

58,963 次观看 • 4 个月前

In 1906, the world-renowned Russian mathematician Andrey A. Markov asked a heretic question of that time: if randomness has memory, do averages still behave or does probability theory collapse? His answer was a new kind of dependence where the next step only remembers the present, P(X_{n+1}=j | X_n=i, X_{n-1}, …) = P_ij, and yet the law of large numbers survives. The path stays noisy, while long-run state frequencies converge to a fixed profile pi = pi P. Markov’s idea is now the workhorse behind estimates by random-walks. MCMC (Markov Chain Monte Carlo) builds a chain whose stationary distribution is the target (posteriors, partition functions, constrained geometries), then uses time-averages as estimates. The same Markov structure shows up in hidden Markov models for time series, in biophysics as channels switching between states, and in control/RL as Markov decision processes. #ProbabilityTheory #Markov #MCMC
0:58

In 1906, the world-renowned Russian mathematician Andrey A. Markov asked a heretic question of that time: if randomness has memory, do averages still behave or does probability theory collapse? His answer was a new kind of dependence where the next step only remembers the present, P(X_{n+1}=j | X_n=i, X_{n-1}, …) = P_ij, and yet the law of large numbers survives. The path stays noisy, while long-run state frequencies converge to a fixed profile pi = pi P. Markov’s idea is now the workhorse behind estimates by random-walks. MCMC (Markov Chain Monte Carlo) builds a chain whose stationary distribution is the target (posteriors, partition functions, constrained geometries), then uses time-averages as estimates. The same Markov structure shows up in hidden Markov models for time series, in biophysics as channels switching between states, and in control/RL as Markov decision processes. #ProbabilityTheory #Markov #MCMC

Mathelirium

288,553 次观看 • 8 个月前

Atomic Truth Before Atoms Had a Name. ⚛️ Long before modern physics, Rishi Kanada described the universe as being made of indivisible particles, anu and paramanu. Matter, motion, time, space, all explained through logic, observation, and natural laws. This wasn’t mythology. This was science that was born in ancient India. ⚛️🇮🇳
0:53

Atomic Truth Before Atoms Had a Name. ⚛️ Long before modern physics, Rishi Kanada described the universe as being made of indivisible particles, anu and paramanu. Matter, motion, time, space, all explained through logic, observation, and natural laws. This wasn’t mythology. This was science that was born in ancient India. ⚛️🇮🇳

JIX5A

14,006 次观看 • 6 个月前

Why the character movement in my custom game engine felt janky and how I fixed it. In a game engine, most often, a character moves using the physics engine. Meaning, the player is not just a coordinate in space but a physical body. It has velocity, it handles collisions, and it interacts with the world. Now, as you might know, physics engines need stability. If you run them at variable framerates, things start breaking. Objects phase through walls or fly off into space because the math becomes unpredictable. This is why most game engines lock their physics loop to a 60Hz fixed rate. But here’s the problem: If you have a high-end system, you don't want to limit it at 60 FPS. That's a waste of good hardware. Now, that said, if the GPU is rendering at 144 FPS but the player's position (physics driven) only updates 60 times a second, it creates a micro-stutter that ruins the "smooth" feel of the game. A good way to fix this is to treat the character as two separate things: 1. The Physics Body (Invisible part): This is the "real" character. It lives in the 60Hz physics world, it moves the player and handles collisions. 2. The Visual Model and Camera (Visible part): This is what the player actually sees. It doesn't care about collisions, its only job is to look nice and smooth at whatever framerate the GPU is pushing. Once you have this separation, you can use interpolation to keep them in sync. Every time the physics clock ticks, you save the previous position of the invisible body before moving it to the new one. Between those ticks, calculate how far we are between the last physics update and the next one. By using this to drive the visible parts of the game, the stutters disappear. The physics loop stays fixed behind the scenes, while the visuals slide smoothly between the snapshots. Example: - Right after a tick: blend_weight= 0.0 (The visual model stays at the old physics position). - Halfway to the next: blend_weight= 0.5 (The visual model slides to the middle point). - Just before the next: blend_weight= 0.9 (The visual model is almost at the new physics position). Pro-Tip A critical mistake I made initially, and one many devs make, is parenting the camera and visible parts directly to the player body. If you do this, the camera inherits the discrete 60Hz physics movement by default. In that setup, interpolation won't work because the camera is "stuck" to the physics clock. For this fix to work you must decouple the camera and visuals from the body and move them separately. Player movement processing in Detis Engine: - fixed_process: Physics runs at 60Hz. Handles collisions and raw movement. - process: Variable rate. Mainly used for player input caching in the player case. - late_process: Variable rate. Handles interpolated camera movement after physics and everything else is done being processed. - render. Submits the final interpolated transforms to the GPU. The test environment in the video is running on an old 2070-based laptop. Hopefully the video compression won't introduce any stutter... I’m sharing this in hopes it helps a fellow dev. Cheers.
0:23

Why the character movement in my custom game engine felt janky and how I fixed it. In a game engine, most often, a character moves using the physics engine. Meaning, the player is not just a coordinate in space but a physical body. It has velocity, it handles collisions, and it interacts with the world. Now, as you might know, physics engines need stability. If you run them at variable framerates, things start breaking. Objects phase through walls or fly off into space because the math becomes unpredictable. This is why most game engines lock their physics loop to a 60Hz fixed rate. But here’s the problem: If you have a high-end system, you don't want to limit it at 60 FPS. That's a waste of good hardware. Now, that said, if the GPU is rendering at 144 FPS but the player's position (physics driven) only updates 60 times a second, it creates a micro-stutter that ruins the "smooth" feel of the game. A good way to fix this is to treat the character as two separate things: 1. The Physics Body (Invisible part): This is the "real" character. It lives in the 60Hz physics world, it moves the player and handles collisions. 2. The Visual Model and Camera (Visible part): This is what the player actually sees. It doesn't care about collisions, its only job is to look nice and smooth at whatever framerate the GPU is pushing. Once you have this separation, you can use interpolation to keep them in sync. Every time the physics clock ticks, you save the previous position of the invisible body before moving it to the new one. Between those ticks, calculate how far we are between the last physics update and the next one. By using this to drive the visible parts of the game, the stutters disappear. The physics loop stays fixed behind the scenes, while the visuals slide smoothly between the snapshots. Example: - Right after a tick: blend_weight= 0.0 (The visual model stays at the old physics position). - Halfway to the next: blend_weight= 0.5 (The visual model slides to the middle point). - Just before the next: blend_weight= 0.9 (The visual model is almost at the new physics position). Pro-Tip A critical mistake I made initially, and one many devs make, is parenting the camera and visible parts directly to the player body. If you do this, the camera inherits the discrete 60Hz physics movement by default. In that setup, interpolation won't work because the camera is "stuck" to the physics clock. For this fix to work you must decouple the camera and visuals from the body and move them separately. Player movement processing in Detis Engine: - fixed_process: Physics runs at 60Hz. Handles collisions and raw movement. - process: Variable rate. Mainly used for player input caching in the player case. - late_process: Variable rate. Handles interpolated camera movement after physics and everything else is done being processed. - render. Submits the final interpolated transforms to the GPU. The test environment in the video is running on an old 2070-based laptop. Hopefully the video compression won't introduce any stutter... I’m sharing this in hopes it helps a fellow dev. Cheers.

Ioannis Koukourakis

48,636 次观看 • 7 个月前

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 次观看 • 2 个月前

ELON: I’D BE SHOCKED IF GROK HASN’T DISCOVERED NEW PHYSICS BY NEXT YEAR “With respect to academic questions, Grok 4 is better than PhD-level in every subject. No exceptions. At times it may lack common sense and it has not yet invented new technologies or discovered new physics, but that is just a matter of time. I think it may discover new technologies as soon as later this year and I would be shocked if it has not done so next year. So I would expect Grok to literally discover new technologies that are actually useful no later than next year and maybe end of this year and it might discover new physics next year and within two years, I'd say almost certainly. Just let that sink in.” Source: Grok 4 Presentation, July 9, 2025
0:50

ELON: I’D BE SHOCKED IF GROK HASN’T DISCOVERED NEW PHYSICS BY NEXT YEAR “With respect to academic questions, Grok 4 is better than PhD-level in every subject. No exceptions. At times it may lack common sense and it has not yet invented new technologies or discovered new physics, but that is just a matter of time. I think it may discover new technologies as soon as later this year and I would be shocked if it has not done so next year. So I would expect Grok to literally discover new technologies that are actually useful no later than next year and maybe end of this year and it might discover new physics next year and within two years, I'd say almost certainly. Just let that sink in.” Source: Grok 4 Presentation, July 9, 2025

Mario Nawfal

1,207,302 次观看 • 1 年前

Jimmy Carr echoing Peter Thiel with a line that lands hard: “Minus the screens from any room, we’re still living in the 1970s. Nothing’s happened in physics since 1972. String theory has not got us anywhere.” But the real shift? “Take the compute power of AI and point it at physics… everything else in science is stamp collecting. Physics is the real thing. That gave us every bit of technology we have. What happens when you point AI at that?” He sees two roads ahead: a world of plenty with 50× productivity and human flourishing… or something that goes “another way.” If AI finally cracks physics in the next decade — what single breakthrough do you hope comes first: limitless energy, gravity control, new materials… or something we haven’t even named yet?
1:03

Jimmy Carr echoing Peter Thiel with a line that lands hard: “Minus the screens from any room, we’re still living in the 1970s. Nothing’s happened in physics since 1972. String theory has not got us anywhere.” But the real shift? “Take the compute power of AI and point it at physics… everything else in science is stamp collecting. Physics is the real thing. That gave us every bit of technology we have. What happens when you point AI at that?” He sees two roads ahead: a world of plenty with 50× productivity and human flourishing… or something that goes “another way.” If AI finally cracks physics in the next decade — what single breakthrough do you hope comes first: limitless energy, gravity control, new materials… or something we haven’t even named yet?

Camus

2,478,570 次观看 • 6 个月前