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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

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25,914 просмотров • 8 месяцев назад •via X (Twitter)

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Mathelirium

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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.

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Mathelirium

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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

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

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Mathelirium

50,548 просмотров • 6 месяцев назад

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 просмотров • 8 месяцев назад

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

147,334 просмотров • 4 месяцев назад

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 просмотров • 3 месяцев назад

A single electron can make a pair of metallic nanostructures glow. As it passes through a metallic bowtie gap, its near field excites surface plasmons, concentrating energy into a tiny hotspot before releasing it as light. This is known as Cathodoluminiscence. It is a process of revealing hidden optical modes at the nanoscale.

A single electron can make a pair of metallic nanostructures glow. As it passes through a metallic bowtie gap, its near field excites surface plasmons, concentrating energy into a tiny hotspot before releasing it as light. This is known as Cathodoluminiscence. It is a process of revealing hidden optical modes at the nanoscale.

Mathelirium

15,174 просмотров • 23 дней назад

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.

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.

Mathelirium

17,580 просмотров • 25 дней назад

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 просмотров • 8 месяцев назад

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 просмотров • 2 месяцев назад

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

Zyphor

122,696 просмотров • 21 дней назад

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁ + a₂) = |a₁|² + |a₂|² + a₁ a₂ + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂ = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,661 просмотров • 8 месяцев назад

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms 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 drop one line that quietly rewires 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) 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. After that, 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. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms 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 drop one line that quietly rewires 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) 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. After that, 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. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 просмотров • 7 месяцев назад

What Curve Beats a Straight Line? Take two fixed points A and B. A bead slides without friction under gravity. Among all possible curves connecting A to B, which one gives the smallest travel time? This is the Brachistochrone problem. The time functional is T[y] = ∫ √((1 + y′²)/(2gy)) dx where y is the vertical drop from the starting point. The winning curve is not a straight line. It is a cycloid x(θ) = a(θ - sin θ) y(θ) = a(1 - cos θ) The bead first falls steeply to gain speed, then uses that speed across the remaining distance. The path is longer than the straight segment, yet the travel time is smaller. The shortest path solves geometry. The fastest path solves variation. #CalculusOfVariations #Brachistochrone #Cycloid #ClassicalMechanics #MathematicalPhysics #Optimization #DifferentialEquations #PhysicsAnimation #Mathematics

What Curve Beats a Straight Line? Take two fixed points A and B. A bead slides without friction under gravity. Among all possible curves connecting A to B, which one gives the smallest travel time? This is the Brachistochrone problem. The time functional is T[y] = ∫ √((1 + y′²)/(2gy)) dx where y is the vertical drop from the starting point. The winning curve is not a straight line. It is a cycloid x(θ) = a(θ - sin θ) y(θ) = a(1 - cos θ) The bead first falls steeply to gain speed, then uses that speed across the remaining distance. The path is longer than the straight segment, yet the travel time is smaller. The shortest path solves geometry. The fastest path solves variation. #CalculusOfVariations #Brachistochrone #Cycloid #ClassicalMechanics #MathematicalPhysics #Optimization #DifferentialEquations #PhysicsAnimation #Mathematics

Mathelirium

17,600 просмотров • 2 месяцев назад

EM Solver Update 02: Tracing where induction heating happens on Earth Space weather changes the electrical currents in and around Earth’s magnetosphere, which changes the magnetic field reaching the planet. SWARM represents that large scale magnetic field using spherical harmonics. The lowest ones are huge and simple, and as you go higher, the patterns get finer and more complicated. Mathematically, there are infinitely many possible modes, but SWARM only needs the first few to describe the large scale magnetospheric field. That’s where the solver comes in.. I take those big magnetic patterns and send them into a much messier Earth: oceans, continents, changing conductivity, and the solid Earth underneath. I’m following that energy into the abyss to see if induction heating can actually explain the abyssal warming. Details:

EM Solver Update 02: Tracing where induction heating happens on Earth Space weather changes the electrical currents in and around Earth’s magnetosphere, which changes the magnetic field reaching the planet. SWARM represents that large scale magnetic field using spherical harmonics. The lowest ones are huge and simple, and as you go higher, the patterns get finer and more complicated. Mathematically, there are infinitely many possible modes, but SWARM only needs the first few to describe the large scale magnetospheric field. That’s where the solver comes in.. I take those big magnetic patterns and send them into a much messier Earth: oceans, continents, changing conductivity, and the solid Earth underneath. I’m following that energy into the abyss to see if induction heating can actually explain the abyssal warming. Details:

Stephanie Buchanan

36,772 просмотров • 7 дней назад

Stokes Phenomenon - When an Invisible Term Switches On In Asymptotic Mathematics, an expression can hide terms that are almost invisible, then suddenly make them appear when you cross a special direction. Here, we visualize the phase function Φ(z; s) = z³/3 - sz, where both z and s are complex numbers. The saddle points are found from Φ′(z; s) = z² - s = 0, and thus the two saddles are z₊ = √s, z₋ = -√s. Near each saddle, the asymptotic contribution looks like exp(Φ(z₊; s)/h) or exp(Φ(z₋; s)/h), where h is a small parameter. The Stokes condition is given by Im(Φ(z₊; s) - Φ(z₋; s)) = 0. This is the direction where the two exponential phases align. Across that line, one saddle contribution switches on or off. The glowing curves are steepest-descent paths. Along them, Im Φ = constant, while Re Φ decreases. This scene is the hidden switchboard of Asymptotic Analysis... two saddles, invisible exponentials, and a glowing Stokes line where a term suddenly becomes real. #Mathematics #Asymptotics #ComplexAnalysis #StokesPhenomenon #MathematicalPhysics #AppliedMathematics #ScientificVisualization #MathArt #STEM

Stokes Phenomenon - When an Invisible Term Switches On In Asymptotic Mathematics, an expression can hide terms that are almost invisible, then suddenly make them appear when you cross a special direction. Here, we visualize the phase function Φ(z; s) = z³/3 - sz, where both z and s are complex numbers. The saddle points are found from Φ′(z; s) = z² - s = 0, and thus the two saddles are z₊ = √s, z₋ = -√s. Near each saddle, the asymptotic contribution looks like exp(Φ(z₊; s)/h) or exp(Φ(z₋; s)/h), where h is a small parameter. The Stokes condition is given by Im(Φ(z₊; s) - Φ(z₋; s)) = 0. This is the direction where the two exponential phases align. Across that line, one saddle contribution switches on or off. The glowing curves are steepest-descent paths. Along them, Im Φ = constant, while Re Φ decreases. This scene is the hidden switchboard of Asymptotic Analysis... two saddles, invisible exponentials, and a glowing Stokes line where a term suddenly becomes real. #Mathematics #Asymptotics #ComplexAnalysis #StokesPhenomenon #MathematicalPhysics #AppliedMathematics #ScientificVisualization #MathArt #STEM

Mathelirium

20,929 просмотров • 2 месяцев назад

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation

Mathelirium

18,616 просмотров • 2 месяцев назад

I took a Jacobi Theta divisor, Θ(z,t) = ∏ⱼ θ₁(z - cⱼ(t) | τ(t)), and let it breathe across a torus, with the small spheres marking its moving zeros... the precise points where Θ(z,t) vanishes.

I took a Jacobi Theta divisor, Θ(z,t) = ∏ⱼ θ₁(z - cⱼ(t) | τ(t)), and let it breathe across a torus, with the small spheres marking its moving zeros... the precise points where Θ(z,t) vanishes.

Mathelirium

42,906 просмотров • 1 месяц назад