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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.

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Mathelirium

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12,451 Aufrufe • vor 3 Tagen •via X (Twitter)

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66,892 Aufrufe • vor 7 Monaten

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
0:30

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 Aufrufe • vor 4 Monaten

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."
8:09

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 Aufrufe • vor 5 Monaten

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.
0:30

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

Mathelirium

29,608 Aufrufe • vor 4 Monaten

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.
0:41

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 Aufrufe • vor 14 Tagen

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
1:00

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 Aufrufe • vor 7 Monaten

A Schrödinger quantum wave packet moves into a double barrier, swells inside the resonant chamber, and then tunnels through.
0:30

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

Mathelirium

25,597 Aufrufe • vor 3 Monaten

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.
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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 Aufrufe • vor 1 Jahr