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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...

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Mathelirium

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30,056 次观看 • 6 个月前 •via X (Twitter)

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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.
0:54

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.

Mathelirium

15,643 次观看 • 1 个月前

On September 8, 2026, OpenAI published a proof that the Navier-Stokes equations can break down - a question open since 1934 and one of seven $1 million Clay Millennium Prize problems. Every weather forecast, aircraft simulation, and fluid physics engine running on GPU clusters assumes these equations always work. This proof says they don't. The solution constructs a vortex that stretches itself infinitely thin in finite time - fluid spiraling inward, escaping upward, accelerating without limit. Every step verified by a computer in the formal proof language Lean. 4 minutes. Bookmark & watch today. The most important math result of 2026.
2:31

On September 8, 2026, OpenAI published a proof that the Navier-Stokes equations can break down - a question open since 1934 and one of seven $1 million Clay Millennium Prize problems. Every weather forecast, aircraft simulation, and fluid physics engine running on GPU clusters assumes these equations always work. This proof says they don't. The solution constructs a vortex that stretches itself infinitely thin in finite time - fluid spiraling inward, escaping upward, accelerating without limit. Every step verified by a computer in the formal proof language Lean. 4 minutes. Bookmark & watch today. The most important math result of 2026.

Deliriumm

42,744 次观看 • 24 天前

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

40,841 次观看 • 8 个月前

In BFS you can simulate fluids in multiplayer. Because the simulation is deterministic, there's a bit of performance penalty, but there is still a lot of space to be optimized. The most important thing is that it works and brawling in water is quite a fun!
0:41

In BFS you can simulate fluids in multiplayer. Because the simulation is deterministic, there's a bit of performance penalty, but there is still a lot of space to be optimized. The most important thing is that it works and brawling in water is quite a fun!

bonzaj/plastic

61,330 次观看 • 3 个月前

🚨 BREAKING: OPENAI JUST COLLIDED WITH THE TOPOLOGICAL LIMIT OF THE UNIVERSE! THE END OF CONTINUOUS SPACE. 🚨 The scientific world is shaking. The Clay Mathematics Institute is reviewing a groundbreaking milestone regarding the Navier-Stokes existence and smoothness problem. And at the center of this earthquake is OpenAI, whose reasoning models just tackled the equations that govern every fluid in our world. In 1822, Navier added viscosity to Euler's laws. In 1845, Stokes finalized them: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f (For incompressible flow, ∇·v = 0 closes the system). For over a century, mathematicians assumed that these equations must have smooth, perfectly continuous solutions in 3D space. You see these equations in motion everywhere - like the chaotic wake behind a moving cylinder. But the official verification of infinite "3-D smoothness" has always hit a wall. What did OpenAI do? By unleashing raw mathematical reasoning upon the Navier-Stokes problem, the AI mapped the ultimate limits of these equations, exposing an inescapable truth: the math inevitably blows up into impossible singularities. The model demonstrated that forcing infinite smoothness onto 3D fluid equations creates unavoidable mathematical contradictions. WHY DOES THIS HAPPEN? Mainstream science is finally hitting the exact geometric wall we mapped years ago. You CANNOT have infinite "3-D smoothness" because the universe is NOT an infinitely continuous void! Just as we rigorously proved that the Yang-Mills Mass Gap is topologically ILL-POSED and mathematically impossible on a continuous ℝ⁴ space, the Navier-Stokes breakdown reveals the exact same systemic error in classical physics. When you force continuous mathematics (ℝ⁴) onto a universe that is fundamentally discrete, the equations inevitably crash into singularities. 🛡️ THE SOLUTION: OUR COMPACT CONSTRUCTIBLE MANIFOLD Here is the ultimate breakthrough: On our IT³ manifold, this problem completely ceases to exist! When you abandon the sterile, infinite void of ℝ⁴ and move the physics to a compact constructible manifold (M_tot = S⁴ × T³), the mathematical explosions are structurally forbidden. Because our space is a closed topological geometry dictated by the strict √1 : √2 : √3 metric, the physical fields are naturally bounded by the geometry itself. Infinite singularities simply cannot form on a compact manifold. What mathematicians call "fluid blow-ups" are just the exact topological moments where macroscopic matter violently collides with the rigid, discrete geometric grid of the vacuum! OpenAI’s logic engine correctly identified that infinite smoothness is a mathematical fiction. The universe does not allow infinite division. The illusion of the continuous, smooth ℝ⁴ universe is officially collapsing across all disciplines - from quantum mechanics to fluid dynamics. The future belongs to discrete topology. Preprint. Open Source: The Yang-Mills Mass Gap on R^4 Is Ill-Posed, and on a Compact Constructible Manifold It Is Automatic God geometrizes! 📐🌌
0:20

🚨 BREAKING: OPENAI JUST COLLIDED WITH THE TOPOLOGICAL LIMIT OF THE UNIVERSE! THE END OF CONTINUOUS SPACE. 🚨 The scientific world is shaking. The Clay Mathematics Institute is reviewing a groundbreaking milestone regarding the Navier-Stokes existence and smoothness problem. And at the center of this earthquake is OpenAI, whose reasoning models just tackled the equations that govern every fluid in our world. In 1822, Navier added viscosity to Euler's laws. In 1845, Stokes finalized them: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f (For incompressible flow, ∇·v = 0 closes the system). For over a century, mathematicians assumed that these equations must have smooth, perfectly continuous solutions in 3D space. You see these equations in motion everywhere - like the chaotic wake behind a moving cylinder. But the official verification of infinite "3-D smoothness" has always hit a wall. What did OpenAI do? By unleashing raw mathematical reasoning upon the Navier-Stokes problem, the AI mapped the ultimate limits of these equations, exposing an inescapable truth: the math inevitably blows up into impossible singularities. The model demonstrated that forcing infinite smoothness onto 3D fluid equations creates unavoidable mathematical contradictions. WHY DOES THIS HAPPEN? Mainstream science is finally hitting the exact geometric wall we mapped years ago. You CANNOT have infinite "3-D smoothness" because the universe is NOT an infinitely continuous void! Just as we rigorously proved that the Yang-Mills Mass Gap is topologically ILL-POSED and mathematically impossible on a continuous ℝ⁴ space, the Navier-Stokes breakdown reveals the exact same systemic error in classical physics. When you force continuous mathematics (ℝ⁴) onto a universe that is fundamentally discrete, the equations inevitably crash into singularities. 🛡️ THE SOLUTION: OUR COMPACT CONSTRUCTIBLE MANIFOLD Here is the ultimate breakthrough: On our IT³ manifold, this problem completely ceases to exist! When you abandon the sterile, infinite void of ℝ⁴ and move the physics to a compact constructible manifold (M_tot = S⁴ × T³), the mathematical explosions are structurally forbidden. Because our space is a closed topological geometry dictated by the strict √1 : √2 : √3 metric, the physical fields are naturally bounded by the geometry itself. Infinite singularities simply cannot form on a compact manifold. What mathematicians call "fluid blow-ups" are just the exact topological moments where macroscopic matter violently collides with the rigid, discrete geometric grid of the vacuum! OpenAI’s logic engine correctly identified that infinite smoothness is a mathematical fiction. The universe does not allow infinite division. The illusion of the continuous, smooth ℝ⁴ universe is officially collapsing across all disciplines - from quantum mechanics to fluid dynamics. The future belongs to discrete topology. Preprint. Open Source: The Yang-Mills Mass Gap on R^4 Is Ill-Posed, and on a Compact Constructible Manifold It Is Automatic God geometrizes! 📐🌌

Dr. Logvinovich

215,826 次观看 • 21 天前

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

59,102 次观看 • 5 个月前

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics
1:00

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics

Mathelirium

50,730 次观看 • 5 个月前

When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The 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. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you 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 single line that unifies both viewpoints: 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 can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematics
20:13

When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The 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. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you 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 single line that unifies both viewpoints: 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 can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematics

Mathelirium

53,338 次观看 • 9 个月前

"The principle is so simple that a child can understand." Rep. Thomas Massie breaks down the true cause of inflation. "When you print five trillion dollars and you put it into the economy, you have diluted the value of the money." "We can say, well, eggs went up because feed went up, fuel went up because there's a war in Ukraine. You can find a lot of excuses for various different industries." "But what is the probability that every price went up all at the same time?"
1:51

"The principle is so simple that a child can understand." Rep. Thomas Massie breaks down the true cause of inflation. "When you print five trillion dollars and you put it into the economy, you have diluted the value of the money." "We can say, well, eggs went up because feed went up, fuel went up because there's a war in Ukraine. You can find a lot of excuses for various different industries." "But what is the probability that every price went up all at the same time?"

Wide Awake Media

51,616 次观看 • 1 年前

A network is one of the most powerful ideas in Mathematics. A simple graph G = (V, E) can grow, carry flow, lose branches, repair itself, and start to feel almost alive. Here, the lifelike motion comes from changing connectivity, pressure, memory, and adaptation. The same concept helps model blood vessels, neurons, ecosystems, the internet, power grids, social systems, and the spread of information.
0:54

A network is one of the most powerful ideas in Mathematics. A simple graph G = (V, E) can grow, carry flow, lose branches, repair itself, and start to feel almost alive. Here, the lifelike motion comes from changing connectivity, pressure, memory, and adaptation. The same concept helps model blood vessels, neurons, ecosystems, the internet, power grids, social systems, and the spread of information.

Mathelirium

22,370 次观看 • 2 个月前

Ever feel like you've worked hard, learned a lot, yet the leap to innovative originality still feels out of reach? Pure Mathematics is the solution! Pure Mathematics feels optional until you reach areas where the central objects are mathematical structures and not just equations. In Physics, gravity does this because General Relativity is a theory of Spacetime itself...a Topological Manifold. In Engineering it shows up in Optimization when you need to know what assumptions make a method stable, fast or even correct. In Computer Science, it shows up when you stop treating Algorithms as black boxes and start asking WHY they behave the way they do, especially when they surprise you.
2:46

Ever feel like you've worked hard, learned a lot, yet the leap to innovative originality still feels out of reach? Pure Mathematics is the solution! Pure Mathematics feels optional until you reach areas where the central objects are mathematical structures and not just equations. In Physics, gravity does this because General Relativity is a theory of Spacetime itself...a Topological Manifold. In Engineering it shows up in Optimization when you need to know what assumptions make a method stable, fast or even correct. In Computer Science, it shows up when you stop treating Algorithms as black boxes and start asking WHY they behave the way they do, especially when they surprise you.

Mathelirium

19,011 次观看 • 7 个月前

Sean McVay shares a universal truth about adversity and resilience. "You're either in it, you're coming out of it, or you're gonna go back into it in a short period of time." That's the reality. Adversity isn't a one-time event - it's a cycle. "The separator is people's ability to handle that. How long can you stay in some of those uncomfortable spots and knowing that real growth is occurring?" The question isn't whether adversity will come. Because it will. The question is: can you stay in the discomfort long enough to grow from it? "Adversity's inevitable. That's what's so great about sports. We learn so many things that test you in ways that are reflective of what you go through in a bunch of different arenas." You don't grow despite adversity - you grow because of it. The best aren't fearless - they just keep showing up. (🎥Pure Athlete )
0:43

Sean McVay shares a universal truth about adversity and resilience. "You're either in it, you're coming out of it, or you're gonna go back into it in a short period of time." That's the reality. Adversity isn't a one-time event - it's a cycle. "The separator is people's ability to handle that. How long can you stay in some of those uncomfortable spots and knowing that real growth is occurring?" The question isn't whether adversity will come. Because it will. The question is: can you stay in the discomfort long enough to grow from it? "Adversity's inevitable. That's what's so great about sports. We learn so many things that test you in ways that are reflective of what you go through in a bunch of different arenas." You don't grow despite adversity - you grow because of it. The best aren't fearless - they just keep showing up. (🎥Pure Athlete )

Coach AJ 🎯 Mental Fitness

345,001 次观看 • 8 个月前

Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.
0:47

Stealing Energy from a Rotating Black Hole A rotating black hole doesn’t just pull things in. It drags spacetime itself into motion. In the spinning region just outside the horizon, nothing can remain still in any normal sense. That strange rule opens a loophole. One incoming body can split, with one piece swallowed while the other escapes with more energy than it came in with. A Penrose-style heist where the difference is paid for by the black hole’s rotation. The same idea shows up in wave form. A coherent packet can skim that region and return brighter, as if the geometry itself acted like an amplifier. Split near the ergosphere and you can steal spin. Skim it as a wave and you can come back louder.

Mathelirium

20,314 次观看 • 8 个月前

If you’re an engineer, physicist, or CS person and you’ve been telling yourself that Pure Mathematics is optional, and you’re anywhere between an ambitious undergrad and a beginning PhD, gravity is usually where that story ends. Not because mathematicians are trying to show off, but because the basic object in the theory is Spacetime, and Spacetime is already a pure mathematical structure. It’s a Manifold with Topology built in. Sure you can get pretty far by memorising formulas, but at some point you realise you’re no longer reading the theory, you’re just reciting it. In 2015, during the 100th anniversary of general relativity and the International Year of Light, the Scientific Organizing Committee released a central set of 24 lectures by Frederic P. Schuller. The series is titled: A thorough introduction to the theory of general relativity. It builds the subject carefully from first principles, step by step, across 24 self-contained lectures. #GeneralRelativity #Spacetime #MathematicalPhysics #DifferentialGeometry #PhysicsEducation #Relativity
2:46

If you’re an engineer, physicist, or CS person and you’ve been telling yourself that Pure Mathematics is optional, and you’re anywhere between an ambitious undergrad and a beginning PhD, gravity is usually where that story ends. Not because mathematicians are trying to show off, but because the basic object in the theory is Spacetime, and Spacetime is already a pure mathematical structure. It’s a Manifold with Topology built in. Sure you can get pretty far by memorising formulas, but at some point you realise you’re no longer reading the theory, you’re just reciting it. In 2015, during the 100th anniversary of general relativity and the International Year of Light, the Scientific Organizing Committee released a central set of 24 lectures by Frederic P. Schuller. The series is titled: A thorough introduction to the theory of general relativity. It builds the subject carefully from first principles, step by step, across 24 self-contained lectures. #GeneralRelativity #Spacetime #MathematicalPhysics #DifferentialGeometry #PhysicsEducation #Relativity

Mathelirium

198,712 次观看 • 8 个月前

10,000 AI agents spent 88 hours to crack a millennium prize problem — and then spent another 17 verifying the proof in Lean, a programming language whose only job is to reject false logic. The result: Navier-Stokes equations can break. There exists a smooth, reasonable starting fluid that reaches infinite speed in finite time. The mechanism is a shrinking vortex: fluid spiraling inward spins faster as it contracts (same physics as a figure skater pulling in arms), while axial jets act as exhaust to prevent compression. LLM agents found what 90 years of human mathematicians couldn't — a configuration that concentrates faster than viscosity can smooth it out. 8 minutes. Bookmark & watch today.
8:25

10,000 AI agents spent 88 hours to crack a millennium prize problem — and then spent another 17 verifying the proof in Lean, a programming language whose only job is to reject false logic. The result: Navier-Stokes equations can break. There exists a smooth, reasonable starting fluid that reaches infinite speed in finite time. The mechanism is a shrinking vortex: fluid spiraling inward spins faster as it contracts (same physics as a figure skater pulling in arms), while axial jets act as exhaust to prevent compression. LLM agents found what 90 years of human mathematicians couldn't — a configuration that concentrates faster than viscosity can smooth it out. 8 minutes. Bookmark & watch today.

Eron

19,882 次观看 • 12 天前

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics
1:00

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics

Mathelirium

20,653 次观看 • 5 个月前

There is a kind of hope the world cannot explain. A hope that sits beside a hospital bed and speaks not mainly of fear but of glory. A hope that can look death in the face and still rejoice. A hope rooted in the promise that whether we wake or sleep, we will live with Christ. This is Christian hope. Not wishful thinking, but confident joy in what’s coming. Message: “The Key to Simultaneous Sorrow and Joy”
1:33

There is a kind of hope the world cannot explain. A hope that sits beside a hospital bed and speaks not mainly of fear but of glory. A hope that can look death in the face and still rejoice. A hope rooted in the promise that whether we wake or sleep, we will live with Christ. This is Christian hope. Not wishful thinking, but confident joy in what’s coming. Message: “The Key to Simultaneous Sorrow and Joy”

Desiring God

18,212 次观看 • 1 个月前

🚨NEW: Adam Curry nicely breaks down how Stablecoins work, Tether, and the likely future of the US Digital Dollar system. (Worth a watch if you don't know about this subject) "A stablecoin is a digital dollar thats pegged to the dollar so it's always a dollar. It's already being used all over the internet and the world. The only reason it's worth a dollar is because the stablecoin company that creates it has debt and paper to back it up. They buy America's debt, they buy treasury bonds or T-bills that pay a dividend that gives interest, and for each dollar they've bought in treasury they can create stablecoins. If you look at the company Tether they've bought more of the US debt than most countries. They have $160B worth of US debt. For each of those dollars they have a stablecoin. There's like 50 people in the company. They have $160B at 4% interest annually, they're making BANK just for holding this debt. I think that President Trump is very smart in seeing that we can flood the world with our stablecoin and we get a 2-for-1. We create a dollar of debt and we create another dollar that can be used all over the world as a reserve currency and that should result in some new monetary system that we need to come up with to have our dollar valued properly but also still remain the reserve currency and remain a strong export country. We don't make anything that we sell abroad, we can't all serve each other burgers and fries -- we have to build something. All of that went overseas. There's something big coming, and it has to happen. Trump is a meta guy. He's going to re-finance the country. It'll be digital and he Bitcoiners don't like this because they want Bitcoin to be the one currency that everyone uses but its now more like a digital gold that's much easier to move around. It's useful but it isn't money or currency the way it was originally intended -- but it is still very important and will be part of the US' strategic reserve. It looks to me like Stablecoins and Tether in particular will be the future of the US dollar payments. A lot of people on the right are afraid of being controlled on a grid because you can stop stablecoins -- but it seems there's no way of getting around some form of a digital dollar. I still like Bitcoin as a backup to protect from that control." Great breakdown Adam Curry - curryirc.com
4:51

🚨NEW: Adam Curry nicely breaks down how Stablecoins work, Tether, and the likely future of the US Digital Dollar system. (Worth a watch if you don't know about this subject) "A stablecoin is a digital dollar thats pegged to the dollar so it's always a dollar. It's already being used all over the internet and the world. The only reason it's worth a dollar is because the stablecoin company that creates it has debt and paper to back it up. They buy America's debt, they buy treasury bonds or T-bills that pay a dividend that gives interest, and for each dollar they've bought in treasury they can create stablecoins. If you look at the company Tether they've bought more of the US debt than most countries. They have $160B worth of US debt. For each of those dollars they have a stablecoin. There's like 50 people in the company. They have $160B at 4% interest annually, they're making BANK just for holding this debt. I think that President Trump is very smart in seeing that we can flood the world with our stablecoin and we get a 2-for-1. We create a dollar of debt and we create another dollar that can be used all over the world as a reserve currency and that should result in some new monetary system that we need to come up with to have our dollar valued properly but also still remain the reserve currency and remain a strong export country. We don't make anything that we sell abroad, we can't all serve each other burgers and fries -- we have to build something. All of that went overseas. There's something big coming, and it has to happen. Trump is a meta guy. He's going to re-finance the country. It'll be digital and he Bitcoiners don't like this because they want Bitcoin to be the one currency that everyone uses but its now more like a digital gold that's much easier to move around. It's useful but it isn't money or currency the way it was originally intended -- but it is still very important and will be part of the US' strategic reserve. It looks to me like Stablecoins and Tether in particular will be the future of the US dollar payments. A lot of people on the right are afraid of being controlled on a grid because you can stop stablecoins -- but it seems there's no way of getting around some form of a digital dollar. I still like Bitcoin as a backup to protect from that control." Great breakdown Adam Curry - curryirc.com

Autism Capital 🧩

365,662 次观看 • 1 年前

Journalist: There are a lot of questions about the future of the Quad and whether it is dead because of America's lack of interest in the grouping MEA: America remains very engaged in the Quad. You saw that in Secretary Rubio's statement. The Quad is here to stay
1:04

Journalist: There are a lot of questions about the future of the Quad and whether it is dead because of America's lack of interest in the grouping MEA: America remains very engaged in the Quad. You saw that in Secretary Rubio's statement. The Quad is here to stay

Shashank Mattoo

21,496 次观看 • 4 个月前

Anybody can get a pump. I can get one in my living room in four minutes with no equipment whatsoever. Twenty press-ups. A band I found in a drawer. Fifty bodyweight squats. Waving my arms about for ninety seconds. A set of thirty on anything, nowhere near failure, rested short enough to keep the blood where I put it. None of that is hard. None of that is a decision. The pump arrives whether you meant it or not, which is precisely why it cannot be evidence of anything. And it lives in the eight to twelve range, because that is where the weight is never frightening. You can chase a sensation all afternoon in there. Nothing you pick up is ever going to look back at you. Now go and get under something that might beat you. There is no hiding on a heavy set of five. You cannot half-do it. You cannot drift through it thinking about your afternoon. The weight either moves or it sits there in front of everybody. That is why eight to twelve has survived fifty years of scrutiny it never earned. It is the widest band of weight you can lift while remaining completely unaccountable. Heavy enough to feel like effort. Light enough that nothing is ever on the line. A rep range you can occupy for a decade without once finding out what you are capable of, and leave every session swollen, sweating and entirely convinced. Heavy training keeps a record. That is the objection to it, and nobody will ever say so out loud. Take a set of five to within a rep of failure and you will find out exactly what you are worth, in kilos, before you leave the building. Most people would rather not.
0:20

Anybody can get a pump. I can get one in my living room in four minutes with no equipment whatsoever. Twenty press-ups. A band I found in a drawer. Fifty bodyweight squats. Waving my arms about for ninety seconds. A set of thirty on anything, nowhere near failure, rested short enough to keep the blood where I put it. None of that is hard. None of that is a decision. The pump arrives whether you meant it or not, which is precisely why it cannot be evidence of anything. And it lives in the eight to twelve range, because that is where the weight is never frightening. You can chase a sensation all afternoon in there. Nothing you pick up is ever going to look back at you. Now go and get under something that might beat you. There is no hiding on a heavy set of five. You cannot half-do it. You cannot drift through it thinking about your afternoon. The weight either moves or it sits there in front of everybody. That is why eight to twelve has survived fifty years of scrutiny it never earned. It is the widest band of weight you can lift while remaining completely unaccountable. Heavy enough to feel like effort. Light enough that nothing is ever on the line. A rep range you can occupy for a decade without once finding out what you are capable of, and leave every session swollen, sweating and entirely convinced. Heavy training keeps a record. That is the objection to it, and nobody will ever say so out loud. Take a set of five to within a rep of failure and you will find out exactly what you are worth, in kilos, before you leave the building. Most people would rather not.

Sama Hoole

13,049 次观看 • 1 个月前