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This scene asks a stranger question : What happens if the material has negative refractive index? As the electron moves through the metamaterial slab, the emitted radiation no longer follows the familiar Cherenkov geometry (see below). The energy flow forms a backward-directed cone, extending opposite to the electron's motion.... show more

Mathelirium

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In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

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The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

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An Electron flies above a metal grating without touching it. As the electron passes, each groove scatters its electromagnetic near field at a slightly different time. The emissions interfere and produce what's called Smith-Purcell radiation. This effect is being explored for tunable Terahertz radiation, biomedical spectroscopy, and high-speed communications.

An Electron flies above a metal grating without touching it. As the electron passes, each groove scatters its electromagnetic near field at a slightly different time. The emissions interfere and produce what's called Smith-Purcell radiation. This effect is being explored for tunable Terahertz radiation, biomedical spectroscopy, and high-speed communications.

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This is a visualization of a Rydberg wavepacket. In simple terms it is a way of showing an electron in a very highly excited quantum state. The Math comes from the Schrödinger equation for hydrogen where the electron is described by a wavefunction rather than as a tiny ball moving on a fixed path. The yellow sphere at the centre is the proton, not the electron. The colored 3D surface around it shows places where the electron is more likely to be, i.e. a probability cloud.

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Google Flow

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