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Will OpenAI and Anthropic ever crack this one? Behind this beautiful turbulence cascade lies a Millennium Prize Problem: For the 3D Navier-Stokes equations, nobody has proved whether every smooth flow remains smooth forever or whether a singularity can form in finite time.

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Mathelirium

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14,805 görüntüleme • 7 gün önce •via X (Twitter)

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A Million-Dollar Storm Mathematics Still Cannot Tame. Turbulence matters because it is the default behavior of real fluids once motion becomes strong enough, e.g. air over wings, blood through vessels, weather in the atmosphere, water in oceans, fuel in engines, and flow in pipes and turbines. We can write down the Navier-Stokes equations for all of this, but in 3D one of the deepest open questions is still whether smooth solutions always stay smooth, or can break down in finite time. That is crucial because it asks whether the basic equations of fluid motion remain mathematically under control forever, or can form a true singularity where the theory itself breaks. Solve that and you got yourself a million dollars
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A Million-Dollar Storm Mathematics Still Cannot Tame. Turbulence matters because it is the default behavior of real fluids once motion becomes strong enough, e.g. air over wings, blood through vessels, weather in the atmosphere, water in oceans, fuel in engines, and flow in pipes and turbines. We can write down the Navier-Stokes equations for all of this, but in 3D one of the deepest open questions is still whether smooth solutions always stay smooth, or can break down in finite time. That is crucial because it asks whether the basic equations of fluid motion remain mathematically under control forever, or can form a true singularity where the theory itself breaks. Solve that and you got yourself a million dollars

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A rocket cuts through a solved Navier-Stokes flow, forcing the fluid to separate around its body and roll into a long turbulent wake. Using the D2Q9 lattice-Boltzmann method, the scene reveals vorticity, strain, Q-criterion vortex cores and wall shear.
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A rocket cuts through a solved Navier-Stokes flow, forcing the fluid to separate around its body and roll into a long turbulent wake. Using the D2Q9 lattice-Boltzmann method, the scene reveals vorticity, strain, Q-criterion vortex cores and wall shear.

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"This tackles all skin complaints and needs in one product, whether you want to blur pores, smooth lines or create a brighter base" - GLAMOUR UK Take the PRIMER quiz to try before you buy:
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"This tackles all skin complaints and needs in one product, whether you want to blur pores, smooth lines or create a brighter base" - GLAMOUR UK Take the PRIMER quiz to try before you buy:

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Tennesseans are raving about the One Big Beautiful Bill. Whether you’re a parent, a small business owner, a server, a farmer, or a senior citizen - this landmark law has something in it for you.
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Tennesseans are raving about the One Big Beautiful Bill. Whether you’re a parent, a small business owner, a server, a farmer, or a senior citizen - this landmark law has something in it for you.

Sen. Marsha Blackburn

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This mesmerizing fountain demonstrates a beautiful combination of laminar flow and total internal reflection, transforming an ordinary stream of water into a living fiber-optic cable Unlike a typical fountain, a laminar nozzle produces an exceptionally smooth, glass-like stream by forcing water to flow in parallel layers with minimal turbulence. The result is a crystal-clear arc that remains remarkably stable as it travels through the air
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This mesmerizing fountain demonstrates a beautiful combination of laminar flow and total internal reflection, transforming an ordinary stream of water into a living fiber-optic cable Unlike a typical fountain, a laminar nozzle produces an exceptionally smooth, glass-like stream by forcing water to flow in parallel layers with minimal turbulence. The result is a crystal-clear arc that remains remarkably stable as it travels through the air

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The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems. To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton, upon whose work Perelman built.
0:21

The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems. To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton, upon whose work Perelman built.

internet archiva

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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.

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🚨NEW: Pam Bondi just DROPPED the hammer on Fox News about the Minneapolis subpoenas! “Whether it’s a public official, whether it’s a law enforcement officer, no one is above the law in this state or in this country, and people will be held accountable”
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🚨NEW: Pam Bondi just DROPPED the hammer on Fox News about the Minneapolis subpoenas! “Whether it’s a public official, whether it’s a law enforcement officer, no one is above the law in this state or in this country, and people will be held accountable”

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It’s now do or die, and we all share the same mindset: to win the America’s Cup. Competing for the America’s Cup is a rare opportunity, a dream few ever get to chase. But for us, that dream has become our mission. The will to win runs deep within every member of this team—through the blood, sweat, and tears we've poured into this journey. This team is a family. We rely on each other every single day, driven by one goal, one vision. We’re going for the win, and the time is now. Whether we face victory or defeat, families and friends stand behind us. Together, as a family, we’re pushing forward. 🇺🇸🪄
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It’s now do or die, and we all share the same mindset: to win the America’s Cup. Competing for the America’s Cup is a rare opportunity, a dream few ever get to chase. But for us, that dream has become our mission. The will to win runs deep within every member of this team—through the blood, sweat, and tears we've poured into this journey. This team is a family. We rely on each other every single day, driven by one goal, one vision. We’re going for the win, and the time is now. Whether we face victory or defeat, families and friends stand behind us. Together, as a family, we’re pushing forward. 🇺🇸🪄

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JUST IN: A truck lost its load this afternoon at the corner of Sam Nujoma Drive and Taal Street in Windhoek, crushing a sedan travelling behind it. It remains unclear at this stage whether there were any injuries or fatalities in the incident. Video: Contributed
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JUST IN: A truck lost its load this afternoon at the corner of Sam Nujoma Drive and Taal Street in Windhoek, crushing a sedan travelling behind it. It remains unclear at this stage whether there were any injuries or fatalities in the incident. Video: Contributed

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Advisor to the Leader & former acting president, Mokhber: No one has the right to cross the red lines of the Leader. The US has proven its bad faith over the past 47 years. As for whether this situation will continue, and whether this 60-day agreement we have will continue or not… a commitment is answered with a commitment. Any commitment they break, we will break as well.
1:06

Advisor to the Leader & former acting president, Mokhber: No one has the right to cross the red lines of the Leader. The US has proven its bad faith over the past 47 years. As for whether this situation will continue, and whether this 60-day agreement we have will continue or not… a commitment is answered with a commitment. Any commitment they break, we will break as well.

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When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics
0:40

When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics

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215 thousand people bro WILL BRAG ABOUT THIS FOR A LONG TIME BECAUSE NOBODY HAS EVER DONE THIS,THE RATEL TAKES THE LEAD AGAIN,GO AND VERIFY #mvoi4all
3:14

215 thousand people bro WILL BRAG ABOUT THIS FOR A LONG TIME BECAUSE NOBODY HAS EVER DONE THIS,THE RATEL TAKES THE LEAD AGAIN,GO AND VERIFY #mvoi4all

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Politician Money Bior has been spotted in Dubai shopping for a supercar. This comes just a few months after he lost the Kasipul Constituency MP seat, where he finished fourth with only 519 votes. It remains unclear whether he will ever return to politics.
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Politician Money Bior has been spotted in Dubai shopping for a supercar. This comes just a few months after he lost the Kasipul Constituency MP seat, where he finished fourth with only 519 votes. It remains unclear whether he will ever return to politics.

The Kenyan Vigilante

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Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #
20:13

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Mathelirium

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Sen. Coons: "Will you commit that no one who has been convicted of assaulting a police officer will receive a payout from this fund?" Blanche: "Anybody can apply…Whether an individual, an Oath Keeper, applies for compensation — anyone in this country can apply."
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Sen. Coons: "Will you commit that no one who has been convicted of assaulting a police officer will receive a payout from this fund?" Blanche: "Anybody can apply…Whether an individual, an Oath Keeper, applies for compensation — anyone in this country can apply."

The Bulwark

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Muslim man in America is asked a simple question, “If you like your religion so much, why don't you go to a Muslim country?” The Muslim man proudly proclaims “This is a Muslim country” and they are here to take over America “whether you like it or not.” He straight up says this is their goal her “This is a Muslim country and every single one of you will be Muslim. It's just that simple because Allah says this religion of mine will enter every single home whether you like it or not. It would overtake every religion whether you like it or not.”
0:20

Muslim man in America is asked a simple question, “If you like your religion so much, why don't you go to a Muslim country?” The Muslim man proudly proclaims “This is a Muslim country” and they are here to take over America “whether you like it or not.” He straight up says this is their goal her “This is a Muslim country and every single one of you will be Muslim. It's just that simple because Allah says this religion of mine will enter every single home whether you like it or not. It would overtake every religion whether you like it or not.”

Wall Street Apes

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Message from #NEIOH ~ This Will Raise Your Frequency "Kindness Will Change The Physiology Of The Body. Hormones And Neurotransmitters Will Flow Freely In Response To This Powerful Choice. Whether Giving A Glass Of Water Or A Large Gift, Kindness Is The Driving Force That Covers The Giver And Receiver With Calm And Beauty." "In That Moment, The Energy Field Closes Tears And Brightens. The Aura Flashes In Lavender, Green And Blue. Every Smile Will Amplify Your Frequency. Just As Every Harsh Word Or Lie Breaks Your Field Into Holes And Openings For Lower Frequencies."
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Message from #NEIOH ~ This Will Raise Your Frequency "Kindness Will Change The Physiology Of The Body. Hormones And Neurotransmitters Will Flow Freely In Response To This Powerful Choice. Whether Giving A Glass Of Water Or A Large Gift, Kindness Is The Driving Force That Covers The Giver And Receiver With Calm And Beauty." "In That Moment, The Energy Field Closes Tears And Brightens. The Aura Flashes In Lavender, Green And Blue. Every Smile Will Amplify Your Frequency. Just As Every Harsh Word Or Lie Breaks Your Field Into Holes And Openings For Lower Frequencies."

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“[The Astros] can redo this thing in 2 or 3 trades and bring in a completely new haul.” #MLBCentral debates whether or not it's too early for the Astros to consider a “reboot”.
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“[The Astros] can redo this thing in 2 or 3 trades and bring in a completely new haul.” #MLBCentral debates whether or not it's too early for the Astros to consider a “reboot”.

MLB Network

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This week Christopher F. Rufo ⚔️ and I ask the big questions about the Iran strikes, and whether this signals a crack-up in the coalition. One central question here, and as a skeptic, one that has required some reflection on my own part is, "Who actually is Trump's base?"
1:25

This week Christopher F. Rufo ⚔️ and I ask the big questions about the Iran strikes, and whether this signals a crack-up in the coalition. One central question here, and as a skeptic, one that has required some reflection on my own part is, "Who actually is Trump's base?"

Lomez

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