正在加载视频...

视频加载失败

加载此视频时出现问题。这可能是由于临时网络问题，或视频可能不可用。

Constructing Weierstrass' function. Continuous, nowhere differentiable, unbounded variation, no intervals of increase or decrease. Fractal dimension 3/2. All things it has in common with mathematical Brownian motion

Almost Sure

8,116 subscribers

11,759 次观看 • 1 年前 •via X (Twitter)

艺术科学技术教育

Anya Rossi• Live Now

Private livecam show

9 条评论

Almost Sure 的头像

Almost Sure1 年前

second time I've written this animation from scratch, as the first was lost when I destroyed my laptop by spilling diet Coke on it, so may as well post it here

Almost Sure 的头像

Almost Sure1 年前

It is Hölder continuous with exponent 1/2. Unlike Brownian motion, which is only locally Hölder continuous for exponents 1/2-ε

FRANK E ELKINS 的头像

FRANK E ELKINS2 年前

“Science doesn’t tell us why the Big Bang happened, how the singularity occurred in the first place, or why it exploded when it did. It tells us there is objective scientific evidence that it occurred. So, what exactly was the Big Bang?” – Book III The Enigmatic Mystery

Sam Power 的头像

Sam Power1 年前

I learned recently that estimating the Hoelder continuity properties of these functions is surprisingly simple (compared to the care that is required for the negative results); really fun stuff!

Almost Sure 的头像

Almost Sure1 年前

indeed (fixed typos and relinked)

cédric bounya 的头像

cédric bounya1 年前

Thanks for the nightmares ! 🫣

Abide By Reason 的头像

Abide By Reason1 年前

this next vid looks like it will be epic!

BIFURCATION{JESUS} 的头像

BIFURCATION{JESUS}1 年前

€_n~ix1+n|¥+|-z{{^am+b-C}

William Selby 的头像

William Selby1 年前

I don't understand it. But it sure is cool!

相关视频

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

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

100,478 次观看 • 5 个月前

Zooming in on Brownian motion:

Zooming in on Brownian motion:

ₕₐₘₚₜₒₙ

67,519 次观看 • 1 年前

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 it became a clean mathematical object. In 1827, Robert Brown watched pollen grains jitter under a microscope and had no microscopic theory to blame; almost a century later, Einstein and Smoluchowski explained that jitter as the visible effect of countless molecular collisions, and then Wiener turned that physical picture into a precise random function of time that now underpins much of modern probability, physics, and quantitative finance. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #MathematicalPhysics #QuantFinance

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 it became a clean mathematical object. In 1827, Robert Brown watched pollen grains jitter under a microscope and had no microscopic theory to blame; almost a century later, Einstein and Smoluchowski explained that jitter as the visible effect of countless molecular collisions, and then Wiener turned that physical picture into a precise random function of time that now underpins much of modern probability, physics, and quantitative finance. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #MathematicalPhysics #QuantFinance

Mathelirium

33,072 次观看 • 5 个月前

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 that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

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 that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

Mathelirium

31,688 次观看 • 3 个月前

$Ribbons of light, endlessly folding. 🌀 In “Fractal Ribbons,” UON VISUALS turns code into motion, revealing a universe built from infinite repetition and variation. 🎶: felix_fractal (Hypnagog) IG$

Ribbons of light, endlessly folding. 🌀 In “Fractal Ribbons,” UON VISUALS turns code into motion, revealing a universe built from infinite repetition and variation. 🎶: felix_fractal (Hypnagog) IG

Sphere

24,441 次观看 • 26 天前

People of Namibia 🇳🇦 The Story of USING Public Money to INCREASE or DECREASE Breasts will no Longer be ALLOWED under our watch. There will be serious consequences of anyone in the Regime using Tax Payers money to Increase/Decrease their Breasts!

People of Namibia 🇳🇦 The Story of USING Public Money to INCREASE or DECREASE Breasts will no Longer be ALLOWED under our watch. There will be serious consequences of anyone in the Regime using Tax Payers money to Increase/Decrease their Breasts!

Prof. Job Shipululo AMUPANDA

25,142 次观看 • 1 年前

Brownian motion is the fundamental martingale that arises in almost all areas of mathematics with a stochastic aspect, used in biology, passing by statistical mechanics and diffusion model.

Brownian motion is the fundamental martingale that arises in almost all areas of mathematics with a stochastic aspect, used in biology, passing by statistical mechanics and diffusion model.

Samuel Vaiter

12,300 次观看 • 1 年前

After all these cases, will the incentive to clear UPSC increase or decrease ? With lone warrior Shiv Aroor 🙏

After all these cases, will the incentive to clear UPSC increase or decrease ? With lone warrior Shiv Aroor 🙏

Shekhar Dutt

17,212 次观看 • 10 个月前

Does the pain increase or decrease?

Does the pain increase or decrease?

BigEyeUG

18,112 次观看 • 3 个月前

Does the pressure increase or decrease?

Does the pressure increase or decrease?

BigEyeUG

86,347 次观看 • 1 年前

dynamic particle system (vertices), circular brownian motion, morph reset (with gsap). #threejs

dynamic particle system (vertices), circular brownian motion, morph reset (with gsap). #threejs

۟

124,062 次观看 • 1 年前

Does the pressure increase or decrease 😂🤣❤️ #3Xtra

Does the pressure increase or decrease 😂🤣❤️ #3Xtra

3Xtra

11,195 次观看 • 7 个月前

“There is nowhere in the Equality Act that says it’s for elite women only. It is for ALL women and girls” “Regardless of where you live or your age, there should be NO males in female sport” 2/3

“There is nowhere in the Equality Act that says it’s for elite women only. It is for ALL women and girls” “Regardless of where you live or your age, there should be NO males in female sport” 2/3

SEENinSport

16,184 次观看 • 1 个月前

Remember: Oil companies are responsible for increase and decrease in Petrol price. “GOVERNMENT” has nothing to do with the oil price.

Remember: Oil companies are responsible for increase and decrease in Petrol price. “GOVERNMENT” has nothing to do with the oil price.

Mahua Moitra Fans

132,149 次观看 • 2 年前

Saboti MP Caleb Amisi word of advice to Gen-Zs. Does the pressure increase or decrease?

Saboti MP Caleb Amisi word of advice to Gen-Zs. Does the pressure increase or decrease?

Cyprian, Is Nyakundi

278,276 次观看 • 1 年前

1/ Create your avatar → Open Midjourney in Discord → Upload 2 selfies → Use the prompt in the video (or a variation of it) → Regenerate until you're happy with your avatar

Moritz Kremb

70,893 次观看 • 3 年前

What metric will you use to Judge Trump's Presidency the most over the next 4 years? - Inflation rate? - Total National Debt increase/decrease? - Unity? - Unemployment rate? - Stock Market? - GDP Increase/decrease? - Number of wars we are involved in? - US Oil Production? - US Manufacturing increase/decrease? - Bitcoin Price? - LGBT Issues? - Number of Political People he locks up? - Number of Government agencies he shuts down or guts? - Other? I'm genuinely curious. Please either reply with your top issue from above, or even rank your top 5.

What metric will you use to Judge Trump's Presidency the most over the next 4 years? - Inflation rate? - Total National Debt increase/decrease? - Unity? - Unemployment rate? - Stock Market? - GDP Increase/decrease? - Number of wars we are involved in? - US Oil Production? - US Manufacturing increase/decrease? - Bitcoin Price? - LGBT Issues? - Number of Political People he locks up? - Number of Government agencies he shuts down or guts? - Other? I'm genuinely curious. Please either reply with your top issue from above, or even rank your top 5.

Brian Krassenstein

35,078 次观看 • 1 年前

Mathematical function using Python

Mathematical function using Python

Python Coding

75,950 次观看 • 1 年前

"MOTION is supposed to belong to the class of things which are continuous; and the INFINITE presents itself first in the continuous" ~ Aristotle, Physics Book III The Primordial Sound 🌀

"MOTION is supposed to belong to the class of things which are continuous; and the INFINITE presents itself first in the continuous" ~ Aristotle, Physics Book III The Primordial Sound 🌀

WhatdoIknow

35,498 次观看 • 8 个月前