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It is remarkable how far Math can go. A graph Laplacian and a non-linear wave equation are enough to produce a cortex-like network that moves, pulses and reorganizes like a living tissue.

Mathelirium

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Mathematics always surprises us with how naturally it can model things that feel alive and real. Here, an abstract graph becomes a living Neural Network, carrying signals through branching connections into a central hypergraph.

Mathematics always surprises us with how naturally it can model things that feel alive and real. Here, an abstract graph becomes a living Neural Network, carrying signals through branching connections into a central hypergraph.

Mathelirium

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What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

Mathelirium

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In Mathematics, a graph is defined as G = (V, E)... a collection of vertices V joined by edges E. From this deceptively simple idea emerges a living neural world, where signals race through a cortical graph and converge upon a morphing hypergraph core, revealing how abstract Mathematics can be transformed into something that appears almost alive.

In Mathematics, a graph is defined as G = (V, E)... a collection of vertices V joined by edges E. From this deceptively simple idea emerges a living neural world, where signals race through a cortical graph and converge upon a morphing hypergraph core, revealing how abstract Mathematics can be transformed into something that appears almost alive.

Mathelirium

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A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.

A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.

Mathelirium

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⚠️ℹ️ Far from dead, some cheat devs behind farmbot tools are just adding “human-like errors” and calling it a day. - Aimbot redesigned with a non-linear progression curve and fake whiffs. - Walkbot movement and aimbot tweaked to feature a non-linear progression curve and human-like inconsistencies. (Like your average solo queue teammate) Tell me again how VAC wiped all cheats off the map. 🎬 From earlier this year, but nothing’s changed.

⚠️ℹ️ Far from dead, some cheat devs behind farmbot tools are just adding “human-like errors” and calling it a day. - Aimbot redesigned with a non-linear progression curve and fake whiffs. - Walkbot movement and aimbot tweaked to feature a non-linear progression curve and human-like inconsistencies. (Like your average solo queue teammate) Tell me again how VAC wiped all cheats off the map. 🎬 From earlier this year, but nothing’s changed.

𝙈𝙖𝙩𝙩.

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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

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What if candlesticks weren't just data points on a timeline but nodes in a living network? Visibility graphs do exactly that. Take a rolling window of price, connect every pair of bars that have unobstructed geometric line of sight to each other, and you get a graph a real mathematical graph with nodes, edges, and topology. Then you stop thinking like a trader and start thinking like a network researcher. Degree distribution tells you how information flows through the price structure. Clustering coefficients reveal whether local patterns form tight communities or fall apart. Betweenness centrality identifies which price bars are structurally critical remove them and the network fragments. These metrics become indicators The conversion algorithm is efficient - there's a 2015 paper by Lan, Mo, Chen, Liu, and Deng that made it fast enough for real-time rolling windows. The Python ecosystem is ready: ts2vg for graph construction, NetworkX for analysis.

What if candlesticks weren't just data points on a timeline but nodes in a living network? Visibility graphs do exactly that. Take a rolling window of price, connect every pair of bars that have unobstructed geometric line of sight to each other, and you get a graph a real mathematical graph with nodes, edges, and topology. Then you stop thinking like a trader and start thinking like a network researcher. Degree distribution tells you how information flows through the price structure. Clustering coefficients reveal whether local patterns form tight communities or fall apart. Betweenness centrality identifies which price bars are structurally critical remove them and the network fragments. These metrics become indicators The conversion algorithm is efficient - there's a 2015 paper by Lan, Mo, Chen, Liu, and Deng that made it fast enough for real-time rolling windows. The Python ecosystem is ready: ts2vg for graph construction, NetworkX for analysis.

gemchanger

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What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

17,459 görüntüleme • 3 ay önce

“A wave is building. The kind of wave that comes once a generation. And people have been pushed too far and they decide all at once, altogether, that enough is enough…” 👀

“A wave is building. The kind of wave that comes once a generation. And people have been pushed too far and they decide all at once, altogether, that enough is enough…” 👀

Maine

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BREAKING: MSDNC is just admitting how real far-left groups Antifa are and there can be NO DENYING the coordinated insurrectionist network that's RAPIDLY compounding in size: Soboroff (reporter): "There is a vast network now of people who are standing up to resist." "When I say that there's a group that is forming to resist [ICE], that includes monitors- people who are watching ICE, Border Patrol, and those federal agents." "I’m going to go to a training later." When will they be treated like the far-left insurrectionist group they have ALWAYS been?

BREAKING: MSDNC is just admitting how real far-left groups Antifa are and there can be NO DENYING the coordinated insurrectionist network that's RAPIDLY compounding in size: Soboroff (reporter): "There is a vast network now of people who are standing up to resist." "When I say that there's a group that is forming to resist [ICE], that includes monitors- people who are watching ICE, Border Patrol, and those federal agents." "I’m going to go to a training later." When will they be treated like the far-left insurrectionist group they have ALWAYS been?

Gunther Eagleman™

19,209 görüntüleme • 9 ay önce

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

47,308 görüntüleme • 7 ay önce

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 görüntüleme • 7 ay önce

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

Mathelirium

15,607 görüntüleme • 4 ay önce

In the 1990s, a remarkable wave of Black sitcoms and sketch comedy reshaped American television. Shows like "In Living Color" and "Living Single" reflected a wide range of Black life and helped broaden how millions of viewers understood Black experiences. That moment, and the long history that helped make it possible, are at the core of Geoff Bennett's new book, "Black Out Loud."

In the 1990s, a remarkable wave of Black sitcoms and sketch comedy reshaped American television. Shows like "In Living Color" and "Living Single" reflected a wide range of Black life and helped broaden how millions of viewers understood Black experiences. That moment, and the long history that helped make it possible, are at the core of Geoff Bennett's new book, "Black Out Loud."

PBS News

80,594 görüntüleme • 5 ay önce

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.

Mathelirium

29,608 görüntüleme • 4 ay önce

A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

Mathelirium

33,258 görüntüleme • 14 gün önce

The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

Mathelirium

31,688 görüntüleme • 7 ay önce

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.

Mathelirium

25,597 görüntüleme • 3 ay önce

MLP by hand✍️Excel ~ I designed this exercise to show how to calculate a four-level network by hand, along with a graph and matching pytorch code. Also, I made a medium and a large version to show how it scales. 👇Join the 'AI Math' community. Download xlsx.

MLP by hand✍️Excel ~ I designed this exercise to show how to calculate a four-level network by hand, along with a graph and matching pytorch code. Also, I made a medium and a large version to show how it scales. 👇Join the 'AI Math' community. Download xlsx.

Tom Yeh

62,253 görüntüleme • 1 yıl önce