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Do you know how we simulate electromagnetic radiation around real systems like antennas, aircraft, and waveguides? We break the geometry into a mesh, a collection of small elements that lets the solver compute Maxwell’s equations locally instead of attacking the whole shape at once. This makes the simulations practical,... show more

Mathelirium

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We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

139,588 views • 8 months ago

We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

76,194 views • 5 months ago

You can use Blender's geometry nodes to solve physics and electromagnetic equations such as Biot-Savart law. You can calculate the magnetic field of a coil anywhere in 3D space and visualise the magnetic fields. Additionally, you can see in real-time how the magnetic field lines change as the coil gets deformed and twisted! #geometrynodes #b3d #blender3d #electronics #Engineering #Science #physics #kicad #circuits

You can use Blender's geometry nodes to solve physics and electromagnetic equations such as Biot-Savart law. You can calculate the magnetic field of a coil anywhere in 3D space and visualise the magnetic fields. Additionally, you can see in real-time how the magnetic field lines change as the coil gets deformed and twisted! #geometrynodes #b3d #blender3d #electronics #Engineering #Science #physics #kicad #circuits

Sam M

543,613 views • 3 years ago

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

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

Mathelirium

30,056 views • 4 months ago

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology

Mathelirium

23,021 views • 6 months ago

A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work such as meshes, stencils, and stability tuning. Once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain.

A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work such as meshes, stencils, and stability tuning. Once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain.

Mathelirium

56,467 views • 4 months ago

AI is a massive unlock for medicine because once you can simulate cells/human biology you can fast track cures for almost all diseases. Precision medicine becomes real. But if we can simulate cells then we are not far off from simulating the world Are we going to be running simulations within simulation? How many simulations deep are we?

AI is a massive unlock for medicine because once you can simulate cells/human biology you can fast track cures for almost all diseases. Precision medicine becomes real. But if we can simulate cells then we are not far off from simulating the world Are we going to be running simulations within simulation? How many simulations deep are we?

Andrew Kang

54,207 views • 1 year ago

“What we need is a giant leap into the future, in a way that makes sense, and a way that addresses the problem and addresses the real world, through real and practical solutions” - COP28 President, Dr Sultan Al Jaber spoke at the #COP28 Transforming Climate Finance Leaders’ Event

“What we need is a giant leap into the future, in a way that makes sense, and a way that addresses the problem and addresses the real world, through real and practical solutions” - COP28 President, Dr Sultan Al Jaber spoke at the #COP28 Transforming Climate Finance Leaders’ Event

COP28 UAE

20,271 views • 2 years ago

Control Systems is the closest thing engineering has to magic! You get to measure what physics is doing right now, then fight back fast enough that the outcome looks impossible. Episode 3 In this scene a tall rocket falls toward a moving barge and lands using only a throttle knob and gimballed engine. We make it unfair on purpose by using crosswind gusts to shove the vehicle. The barge drifts and gently tilts so the target keeps sliding. Actuator lag and hard limits clamp the gimbal and throttle. A slosh-like internal wobble kicks in late and tries to phase-lag the controller into oscillations. You see the controller ride all these.

Control Systems is the closest thing engineering has to magic! You get to measure what physics is doing right now, then fight back fast enough that the outcome looks impossible. Episode 3 In this scene a tall rocket falls toward a moving barge and lands using only a throttle knob and gimballed engine. We make it unfair on purpose by using crosswind gusts to shove the vehicle. The barge drifts and gently tilts so the target keeps sliding. Actuator lag and hard limits clamp the gimbal and throttle. A slosh-like internal wobble kicks in late and tries to phase-lag the controller into oscillations. You see the controller ride all these.

Mathelirium

22,696 views • 5 months ago

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 #

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

40,739 views • 5 months ago

Faster Than Light Without Moving Faster Than Light In 1994, Miguel Alcubierre discovered a solution to Einstein’s Field Equations where a region of Spacetime can effectively travel faster than light without anything locally breaking Relativity. The object inside the bubble never outruns light in its own local frame. Instead, Spacetime contracts ahead of the bubble, expands behind it, and carries the interior forward. The motion comes from geometry itself.

Faster Than Light Without Moving Faster Than Light In 1994, Miguel Alcubierre discovered a solution to Einstein’s Field Equations where a region of Spacetime can effectively travel faster than light without anything locally breaking Relativity. The object inside the bubble never outruns light in its own local frame. Instead, Spacetime contracts ahead of the bubble, expands behind it, and carries the interior forward. The motion comes from geometry itself.

Mathelirium

17,688 views • 2 months ago

This guy isn’t real. Brutal performance against the most inform attack in the world at that time. The greatest center-defender prodigy in modern-history, how do you remodel the whole theory of attacking and defending at 17?

This guy isn’t real. Brutal performance against the most inform attack in the world at that time. The greatest center-defender prodigy in modern-history, how do you remodel the whole theory of attacking and defending at 17?

Roby

74,087 views • 6 months ago

Of all professions, electrical engineers are the ones that impress me the most. It’s not that they know more math. It’s how naturally they use it. A lot of ideas that sit in the pure-math neighborhood end up powering things like cryptography, coding theory, and information theory. I’d always known that in theory. What shocked me was seeing the same ideas running real systems: probability steering decisions in digital comms, optimisation shaping hardware, and information theory acting like a hard constraint on what’s even possible. Working with them on research was humbling. It made me feel like I knew nothing, in the best way. It also made me rethink what being good at math means. In my optimisation course, Space Mapping was one of the concepts that really stuck with me: you keep a computationally cheap coarse model f_c(x) that runs fast but lies, a brutally expensive fine model f_f(x) that tells the truth, and you iteratively adjust a mapping T so that f_c(T(x)) shadows f_f(x) where it matters. You do almost all the optimisation on the cheap side and call the fine model only sparingly. It’s a very engineer move: admit the model is wrong, then make it useful anyway. John Bandler, a Canadian engineer and professor, formalised this in the early 1990s and showed you could make full-wave electromagnetic optimisation practical rather than masochistic. He founded Optimization Systems Associates in 1983 to commercialise the idea, and in 1997 Hewlett-Packard bought the company and folded its tools into what became HP EEsof, then Agilent, now Keysight’s RF design stack. #SpaceMapping #ComputationalElectromagnetics #RFDesign #NonLinearOptimization #AntennaDesign

Of all professions, electrical engineers are the ones that impress me the most. It’s not that they know more math. It’s how naturally they use it. A lot of ideas that sit in the pure-math neighborhood end up powering things like cryptography, coding theory, and information theory. I’d always known that in theory. What shocked me was seeing the same ideas running real systems: probability steering decisions in digital comms, optimisation shaping hardware, and information theory acting like a hard constraint on what’s even possible. Working with them on research was humbling. It made me feel like I knew nothing, in the best way. It also made me rethink what being good at math means. In my optimisation course, Space Mapping was one of the concepts that really stuck with me: you keep a computationally cheap coarse model f_c(x) that runs fast but lies, a brutally expensive fine model f_f(x) that tells the truth, and you iteratively adjust a mapping T so that f_c(T(x)) shadows f_f(x) where it matters. You do almost all the optimisation on the cheap side and call the fine model only sparingly. It’s a very engineer move: admit the model is wrong, then make it useful anyway. John Bandler, a Canadian engineer and professor, formalised this in the early 1990s and showed you could make full-wave electromagnetic optimisation practical rather than masochistic. He founded Optimization Systems Associates in 1983 to commercialise the idea, and in 1997 Hewlett-Packard bought the company and folded its tools into what became HP EEsof, then Agilent, now Keysight’s RF design stack. #SpaceMapping #ComputationalElectromagnetics #RFDesign #NonLinearOptimization #AntennaDesign

Mathelirium

59,165 views • 6 months ago

The ultimate Engineering flex is making almost any shape fly.😁 AUREOLE - PVRA TriDisc is our experimental drone concept that replaces spinning propellers with Perimetric Vectoring Ring Arrays (PVRA). We place micro-nozzle belts around tri-lobed ring. These tiny jets push air to generate lift, lateral motion, and rotational control for the craft. This scene is a proof-of-concept rigid body dynamics demo rather than a full CFD. We juste wanted to see if the shape can fly. But everything is still grounded in the language of flight...control systems, aerodynamics, physics and mathematics.

The ultimate Engineering flex is making almost any shape fly.😁 AUREOLE - PVRA TriDisc is our experimental drone concept that replaces spinning propellers with Perimetric Vectoring Ring Arrays (PVRA). We place micro-nozzle belts around tri-lobed ring. These tiny jets push air to generate lift, lateral motion, and rotational control for the craft. This scene is a proof-of-concept rigid body dynamics demo rather than a full CFD. We juste wanted to see if the shape can fly. But everything is still grounded in the language of flight...control systems, aerodynamics, physics and mathematics.

Mathelirium

44,691 views • 4 months ago

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

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

50,544 views • 3 months ago

I really want to show off the cutting edge tech the team is building and how we're turning AI image generation into real living environments using gaussian splats and at the :19 mark of this video you can see how we're blending splats together to create a real world. What this looks like from a gaming perspective is this all happens behind the players view to simulate a real world environment. As you move through the space, distance becomes a variable to consider on how images update and are experienced. We are creating a new reality in real time.

I really want to show off the cutting edge tech the team is building and how we're turning AI image generation into real living environments using gaussian splats and at the :19 mark of this video you can see how we're blending splats together to create a real world. What this looks like from a gaming perspective is this all happens behind the players view to simulate a real world environment. As you move through the space, distance becomes a variable to consider on how images update and are experienced. We are creating a new reality in real time.

𝔹𝕃𝕍ℂ𝕂𝕃!𝔾ℍ𝕋

25,084 views • 1 month ago

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.

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 views • 4 months ago

Code and data are now online for CameraHMR, our state-of-the-art parametric 3D human pose and shape (HPS) estimation method that will appear at hashtag#3DV2025. There are 4 key contributions that make it so accurate and robust: 1. To get accurate 3D shape and pose as well as good alignment to image features, you need to know the focal length of the camera. To solve this, we train HumanFOV to compute the field of view. 2. We introduce CameraHMR, which integrates HumanFOV into HMR2.0 to exploit the estimated focal length. 3. To get accurate pseudo ground truth (pGT) training data, we compute the focal length for images in 4DHumans dataset and modify SMPLify to take this into account. 4. But SMPLify only uses sparse 2D keypoints, which do not capture body shape. So we train a dense surface keypoint detector, DenseKP, on BEDLAM and run it on 4DHumans, resulting in improved body shape. The resulting method is CamSMPLify. We iterate training CameraHMR and running CamSMPLify on the training set initialized with CameraHMR. This results in much improved pGT for 4DHumans and a SOTA single-image HMR method.

Code and data are now online for CameraHMR, our state-of-the-art parametric 3D human pose and shape (HPS) estimation method that will appear at hashtag#3DV2025. There are 4 key contributions that make it so accurate and robust: 1. To get accurate 3D shape and pose as well as good alignment to image features, you need to know the focal length of the camera. To solve this, we train HumanFOV to compute the field of view. 2. We introduce CameraHMR, which integrates HumanFOV into HMR2.0 to exploit the estimated focal length. 3. To get accurate pseudo ground truth (pGT) training data, we compute the focal length for images in 4DHumans dataset and modify SMPLify to take this into account. 4. But SMPLify only uses sparse 2D keypoints, which do not capture body shape. So we train a dense surface keypoint detector, DenseKP, on BEDLAM and run it on 4DHumans, resulting in improved body shape. The resulting method is CamSMPLify. We iterate training CameraHMR and running CamSMPLify on the training set initialized with CameraHMR. This results in much improved pGT for 4DHumans and a SOTA single-image HMR method.

Michael Black

21,696 views • 1 year ago

Buttigieg: Everyday life is better in a country where the people get to tell the government what to do instead of the other way around. You can make everyday life better in this state. You can get a problem solver into the highest office in the Commonwealth, and send a message of hope that will be noticed around the country

Buttigieg: Everyday life is better in a country where the people get to tell the government what to do instead of the other way around. You can make everyday life better in this state. You can get a problem solver into the highest office in the Commonwealth, and send a message of hope that will be noticed around the country

FactPost

38,726 views • 9 months ago

The world we live in is run on BLACKMAIL, LIES, and CORRUPTION. A wealthy elite group of families from a specific group of bloodlines sit at the top of this global power structure. They use children as blackmail tools for their operations and occult belief systems to control governments and people in high places all around the world. It's time we come together, set aside our differences and put our time and energy into defeating the real enemy of We, The People.

The world we live in is run on BLACKMAIL, LIES, and CORRUPTION. A wealthy elite group of families from a specific group of bloodlines sit at the top of this global power structure. They use children as blackmail tools for their operations and occult belief systems to control governments and people in high places all around the world. It's time we come together, set aside our differences and put our time and energy into defeating the real enemy of We, The People.

The SCIF

225,619 views • 8 months ago