正在加载视频...

视频加载失败

加载此视频时出现问题。这可能是由于临时网络问题，或视频可能不可用。

Spherical Harmonics 🤝 Fourier Transform I’m modeling the cell surface as a time-varying field u(θ,φ). Budding is energy shifting into higher angular frequencies. Decompose u into spherical harmonics (the Fourier basis on a sphere), evolve the coefficients aₗₘ(t) where growth/smoothing is clean, then inverse-transform back into a breathing, budding... show more

Mathelirium

35,639 subscribers

25,881 次观看 • 6 个月前 •via X (Twitter)

Anya Rossi• Live Now

Private livecam show

0 条评论

暂无评论

原始帖子的评论将显示在这里

相关视频

Fourier Transform 🤝 Spherical Harmonics = Viral Budding Geometry Viral Budding of an Infected Cell

Fourier Transform 🤝 Spherical Harmonics = Viral Budding Geometry Viral Budding of an Infected Cell

Mathelirium

11,817 次观看 • 4 个月前

This video presents a recent breakthrough called the Sparse Fourier Transform (SFT). This algorithm yields an exponential speed-up over the celebrated Fast Fourier Transform (FFT) when asked to extract a small number of dominant Fourier coefficients.

This video presents a recent breakthrough called the Sparse Fourier Transform (SFT). This algorithm yields an exponential speed-up over the celebrated Fast Fourier Transform (FFT) when asked to extract a small number of dominant Fourier coefficients.

Mathelirium

50,777 次观看 • 4 个月前

Nuclear Vibrations via Spherical Harmonics

Nuclear Vibrations via Spherical Harmonics

Mathelirium

16,274 次观看 • 4 个月前

A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

Mathelirium

34,713 次观看 • 2 个月前

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

Hans Chiu

220,123 次观看 • 1 年前

The Fast Fourier Transform is the most important numerical algorithm of our lifetime - Gilbert Strang, MIT

The Fast Fourier Transform is the most important numerical algorithm of our lifetime - Gilbert Strang, MIT

Mathelirium

50,548 次观看 • 3 个月前

Have you seen anything more beautiful today? Fourier Modes Breathing Life Into a Torus

Have you seen anything more beautiful today? Fourier Modes Breathing Life Into a Torus

Mathelirium

12,751 次观看 • 5 个月前

A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the derivative operator written in Fourier language u(x) -> U(ξ) -> iξU(ξ) -> ∂u/∂x Then the second lens brings the field back to the real space. What comes out is no longer just a focused beam. It is the spatial derivative of the input field, computed by light as it propagates. So, this is the serious promise of optical computing. A physical optical train can perform operations that usually live inside numerical code: differentiation, filtering, convolution, edge detection, correlation, and many other linear transforms.

A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the derivative operator written in Fourier language u(x) -> U(ξ) -> iξU(ξ) -> ∂u/∂x Then the second lens brings the field back to the real space. What comes out is no longer just a focused beam. It is the spatial derivative of the input field, computed by light as it propagates. So, this is the serious promise of optical computing. A physical optical train can perform operations that usually live inside numerical code: differentiation, filtering, convolution, edge detection, correlation, and many other linear transforms.

Mathelirium

146,719 次观看 • 2 个月前

Here's one but with the Fourier transform settings a bit wonky

Here's one but with the Fourier transform settings a bit wonky

Sauers

128,866 次观看 • 1 年前

The Fourier series, modeling periodic functions with cosines and sines, and computing Fourier coefficients as projections onto orthogonal bases in Python. source: Steve Brunton

The Fourier series, modeling periodic functions with cosines and sines, and computing Fourier coefficients as projections onto orthogonal bases in Python. source: Steve Brunton

tetsuo

80,704 次观看 • 8 个月前

On Meta Quest you can now transform any flat surface into a virtual keyboard. We truly live in the future.

On Meta Quest you can now transform any flat surface into a virtual keyboard. We truly live in the future.

Nathie 🔜 AWE

55,960 次观看 • 5 个月前

Fourier transform of a mechanical music box shows up the imprinted pattern 🎶

Fourier transform of a mechanical music box shows up the imprinted pattern 🎶

Jithin B.P., Ph.D.

82,933 次观看 • 5 个月前

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 次观看 • 26 天前

MIT engineers have developed an ultralight fabric solar cell that can transform virtually any surface into a power source. Here’s how they work.

MIT engineers have developed an ultralight fabric solar cell that can transform virtually any surface into a power source. Here’s how they work.

Interesting Engineering

54,248 次观看 • 1 年前

One of my favorite features on Apple Vision Pro is the ability to transform old photos into 3D memories. It's like creating a window into the past.

One of my favorite features on Apple Vision Pro is the ability to transform old photos into 3D memories. It's like creating a window into the past.

Nathie 🔜 AWE

13,994 次观看 • 2 年前

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

66,204 次观看 • 5 个月前

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Mathelirium

22,170 次观看 • 14 天前

The only way to safeguard our sovereignty is to accelerate responsible resource development. This is Ontario's plan to transform Canada into a self-reliant clean energy superpower.

The only way to safeguard our sovereignty is to accelerate responsible resource development. This is Ontario's plan to transform Canada into a self-reliant clean energy superpower.

Stephen Lecce

10,388 次观看 • 1 年前

This is the BEST way to digitize your math notes. 🤯 Grok 4 converted a Fourier Series approximation for f(x)=x^2 into clean, formatted LaTeX!

This is the BEST way to digitize your math notes. 🤯 Grok 4 converted a Fourier Series approximation for f(x)=x^2 into clean, formatted LaTeX!

tetsuo

115,562 次观看 • 8 个月前