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A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking the PDE residual straight into the loss. They came out of a very practical pain point.... show more

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Physics-Informed Neural Operators: Learning The Solver, Not Just One Solution Our PINN scene learned one solution field for one PDE setup. A Physics-Informed Neural Operator learns the map from input fields, like material coefficients or source terms, to the full solution across a whole family of PDE problems. So, the goal is no longer just one approximate answer, but a reusable solver-shaped object guided by the physics itself.

Physics-Informed Neural Operators: Learning The Solver, Not Just One Solution Our PINN scene learned one solution field for one PDE setup. A Physics-Informed Neural Operator learns the map from input fields, like material coefficients or source terms, to the full solution across a whole family of PDE problems. So, the goal is no longer just one approximate answer, but a reusable solver-shaped object guided by the physics itself.

Mathelirium

25,935 görüntüleme • 2 ay önce

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.

Harper Carroll

26,930 görüntüleme • 7 ay önce

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Mathelirium

40,739 görüntüleme • 4 ay önce

Welcome to Caitlin of Blacksmithing. ⚒️ “A lot of folks come into the shop and they think, ‘Oh blacksmithing! I’ll be great at this,’” Caitlin Morris says. “About 15 minutes into that process of hitting things as hard as they possibly can, they start to fatigue.” Often, students mistakenly think that, to be a good blacksmith, you need to hit things as hard as you can. In reality, using a hammer should be as easy as walking – if you understand your body. Once students learn how to use a hammer, they experience the magic of taking tiny pieces of scrap metal and watching them turn into something beautiful. For Caitlin, blacksmithing is an entry point to understanding humanity.

Welcome to Caitlin of Blacksmithing. ⚒️ “A lot of folks come into the shop and they think, ‘Oh blacksmithing! I’ll be great at this,’” Caitlin Morris says. “About 15 minutes into that process of hitting things as hard as they possibly can, they start to fatigue.” Often, students mistakenly think that, to be a good blacksmith, you need to hit things as hard as you can. In reality, using a hammer should be as easy as walking – if you understand your body. Once students learn how to use a hammer, they experience the magic of taking tiny pieces of scrap metal and watching them turn into something beautiful. For Caitlin, blacksmithing is an entry point to understanding humanity.

Washington Post Opinions

40,851 görüntüleme • 10 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

66,204 görüntüleme • 5 ay önce

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

23,205 görüntüleme • 19 gün önce

Ever feel like you've worked hard, learned a lot, yet the leap to innovative originality still feels out of reach? Pure Mathematics is the solution! Pure Mathematics feels optional until you reach areas where the central objects are mathematical structures and not just equations. In Physics, gravity does this because General Relativity is a theory of Spacetime itself...a Topological Manifold. In Engineering it shows up in Optimization when you need to know what assumptions make a method stable, fast or even correct. In Computer Science, it shows up when you stop treating Algorithms as black boxes and start asking WHY they behave the way they do, especially when they surprise you.

Ever feel like you've worked hard, learned a lot, yet the leap to innovative originality still feels out of reach? Pure Mathematics is the solution! Pure Mathematics feels optional until you reach areas where the central objects are mathematical structures and not just equations. In Physics, gravity does this because General Relativity is a theory of Spacetime itself...a Topological Manifold. In Engineering it shows up in Optimization when you need to know what assumptions make a method stable, fast or even correct. In Computer Science, it shows up when you stop treating Algorithms as black boxes and start asking WHY they behave the way they do, especially when they surprise you.

Mathelirium

19,011 görüntüleme • 3 ay önce

"If you have ambitions to be a Professor at Yale, Harvard, or one of these big institutions, the way you do that is mostly by getting funding for research. It's by getting a lot of publishing - a lot flying around the world to talk about something. You need all of these things to bolster your reputation as an international expert in this area. Now, the best way to do that, at least in psychiatry but I suspect in other areas of medicine, is actually to collaborate with pharmaceutical companies to run clinical trials. They write the publications for you - most of it, they do a lot of the grunt work, they give you the protocols, and then, essentially, they fund you and fly you around the place. I used to remember, as a junior doctor, looking up to the Professors. And they would say, 'These drugs are safe and effective. There's not a lot of problems there.' ...What I've seen more and more is that they've been compromised, and that they have incentives tied to the money and to career advancement. That means that they can't be as forthright as they ought to be about these things. And it's gotten to a problem where these are the Professors who are training the doctors who are setting the tone for what is acceptable care out there. And I think that's one of the biggest problems out there." - @taperclinic

"If you have ambitions to be a Professor at Yale, Harvard, or one of these big institutions, the way you do that is mostly by getting funding for research. It's by getting a lot of publishing - a lot flying around the world to talk about something. You need all of these things to bolster your reputation as an international expert in this area. Now, the best way to do that, at least in psychiatry but I suspect in other areas of medicine, is actually to collaborate with pharmaceutical companies to run clinical trials. They write the publications for you - most of it, they do a lot of the grunt work, they give you the protocols, and then, essentially, they fund you and fly you around the place. I used to remember, as a junior doctor, looking up to the Professors. And they would say, 'These drugs are safe and effective. There's not a lot of problems there.' ...What I've seen more and more is that they've been compromised, and that they have incentives tied to the money and to career advancement. That means that they can't be as forthright as they ought to be about these things. And it's gotten to a problem where these are the Professors who are training the doctors who are setting the tone for what is acceptable care out there. And I think that's one of the biggest problems out there." - @taperclinic

Jan Jekielek

43,499 görüntüleme • 1 yıl önce

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,340 görüntüleme • 2 ay önce

The WEF's solution to the ecological crisis is the problem itself: the complete financialization of nature. This WEF panelist says the flaw is that nature is treated as "free" and "unlimited." Their solution? Not stewardship, but accounting. Not respect, but valuation. They openly argue to "bring nature onto the balance sheet." This isn't a metaphor. It's a corporatist fantasy to transform the living world into a derivative—a quantifiable, tradeable, and ultimately owned asset class for the very institutions that destroyed it. This is the ultimate enclosure of the global commons. They see a forest not as an ecosystem, but as undervalued carbon sequestration credits. A watershed as a utility to be packaged and sold. Their goal isn't to save a living planet. It's to create a profitable, administrable simulation of one. They can't imagine a world without a balance sheet, so they plan to put the entire planet on one. This is the dystopian endgame of stakeholder capitalism. Resistance is not just environmental; it is a defense of the very concept of the commons.

The WEF's solution to the ecological crisis is the problem itself: the complete financialization of nature. This WEF panelist says the flaw is that nature is treated as "free" and "unlimited." Their solution? Not stewardship, but accounting. Not respect, but valuation. They openly argue to "bring nature onto the balance sheet." This isn't a metaphor. It's a corporatist fantasy to transform the living world into a derivative—a quantifiable, tradeable, and ultimately owned asset class for the very institutions that destroyed it. This is the ultimate enclosure of the global commons. They see a forest not as an ecosystem, but as undervalued carbon sequestration credits. A watershed as a utility to be packaged and sold. Their goal isn't to save a living planet. It's to create a profitable, administrable simulation of one. They can't imagine a world without a balance sheet, so they plan to put the entire planet on one. This is the dystopian endgame of stakeholder capitalism. Resistance is not just environmental; it is a defense of the very concept of the commons.

Camus

83,269 görüntüleme • 9 ay önce

Bill Maher's panelists have come to the conclusion that the Democrat party has been branded as the party of unnecessary regulations: Ezra Klein: "I want regulation." Bill Maher: "If you have 400000 regulations, and you can't build a high speed rail, you need someone to come in here." Ezra Klein: "We need a new word: You say 'deregulate' and a lot of liberals and leftists shut down. It's such a right coded word. Sometimes you need to deregulate the market. Change the word to rules: Some rules are good, some rules are bad. You want to get rid of bad rules. The things that liberals regulate more is the government itself. Democrats do need to learn something: Not lawlessness, but kind of a relentlessness of what you're trying to achieve." 'Democrats do need to learn something:' They need to learn how to admit they were wrong.

Bill Maher's panelists have come to the conclusion that the Democrat party has been branded as the party of unnecessary regulations: Ezra Klein: "I want regulation." Bill Maher: "If you have 400000 regulations, and you can't build a high speed rail, you need someone to come in here." Ezra Klein: "We need a new word: You say 'deregulate' and a lot of liberals and leftists shut down. It's such a right coded word. Sometimes you need to deregulate the market. Change the word to rules: Some rules are good, some rules are bad. You want to get rid of bad rules. The things that liberals regulate more is the government itself. Democrats do need to learn something: Not lawlessness, but kind of a relentlessness of what you're trying to achieve." 'Democrats do need to learn something:' They need to learn how to admit they were wrong.

Eric Abbenante

465,521 görüntüleme • 1 yıl önce

Creativity, logic and objectivity makes you a scientist, a better one is when you are able to problem solve the physics of sound. Musicians are one of the finest aptitudes known to man❤️

Creativity, logic and objectivity makes you a scientist, a better one is when you are able to problem solve the physics of sound. Musicians are one of the finest aptitudes known to man❤️

PRINCE KAYBEE

11,939 görüntüleme • 5 ay önce

Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working the way we expect. Not just numerically different but qualitatively different. The issue isn’t the force law by itself but it's geometry. Gauss’s law ties inverse-square forces to the surface area of spheres, and sphere geometry depends on dimension. Change the dimension, and the same-looking force produces a different potential, a different balance of attraction and inertia, and a different fate for motion.

Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working the way we expect. Not just numerically different but qualitatively different. The issue isn’t the force law by itself but it's geometry. Gauss’s law ties inverse-square forces to the surface area of spheres, and sphere geometry depends on dimension. Change the dimension, and the same-looking force produces a different potential, a different balance of attraction and inertia, and a different fate for motion.

Mathelirium

30,393 görüntüleme • 2 ay önce

Elon Musk just stripped away every emotional narrative around paralysis and reduced it to a pure engineering equation. The human nervous system is not mystical. It is a biological wiring grid. When a wire breaks, you build a bypass. Traditional medicine treats a severed spinal cord as a permanent biological endpoint. Musk treats it as a broken routing switch. Musk: “It’s basically a communications bridge. You bridge the communications from the motor cortex past the point in the neck or spine where the nerves are damaged.” Not a miracle. Not a mystery. A bridge. Musk: “It is possible from a physics standpoint to restore full body functionality. There is nothing that prevents it happening from a physics standpoint.” The physics already check out. This is not a question of possibility. It is a question of execution speed. We are building AGI by mastering computational physics in silicon. Neuralink is applying that same mastery to carbon. The human body is not a sacred text. It is a machine. And machines can be patched. But this is bigger than medicine. Humanity’s ability to interface with superintelligence is currently bottlenecked by thumbs typing on a glass screen. Neuralink is the solution to that constraint dressed as a medical device. If you can bridge the brain past a broken spine, you can bridge the brain to a data center. Healing the paralyzed is step one. Merging with superintelligence is the endgame. Musk: “It’s a very hard technical problem, right, but there is nothing that prevents it happening from a physics standpoint.” Somewhere right now, a person is sitting in a wheelchair. An engineer is sitting in a lab. Neither knows the other exists. But one of them is quietly rewriting the definition of permanent. And it isn’t the one in the wheelchair.

Elon Musk just stripped away every emotional narrative around paralysis and reduced it to a pure engineering equation. The human nervous system is not mystical. It is a biological wiring grid. When a wire breaks, you build a bypass. Traditional medicine treats a severed spinal cord as a permanent biological endpoint. Musk treats it as a broken routing switch. Musk: “It’s basically a communications bridge. You bridge the communications from the motor cortex past the point in the neck or spine where the nerves are damaged.” Not a miracle. Not a mystery. A bridge. Musk: “It is possible from a physics standpoint to restore full body functionality. There is nothing that prevents it happening from a physics standpoint.” The physics already check out. This is not a question of possibility. It is a question of execution speed. We are building AGI by mastering computational physics in silicon. Neuralink is applying that same mastery to carbon. The human body is not a sacred text. It is a machine. And machines can be patched. But this is bigger than medicine. Humanity’s ability to interface with superintelligence is currently bottlenecked by thumbs typing on a glass screen. Neuralink is the solution to that constraint dressed as a medical device. If you can bridge the brain past a broken spine, you can bridge the brain to a data center. Healing the paralyzed is step one. Merging with superintelligence is the endgame. Musk: “It’s a very hard technical problem, right, but there is nothing that prevents it happening from a physics standpoint.” Somewhere right now, a person is sitting in a wheelchair. An engineer is sitting in a lab. Neither knows the other exists. But one of them is quietly rewriting the definition of permanent. And it isn’t the one in the wheelchair.

Dustin

508,681 görüntüleme • 3 ay önce

Do you know how we simulate electromagnetic radiation around real systems like antennas, aircraft, and waveguides? We break the geometry into a mesh, a collection of small elements that lets the solver compute Maxwell’s equations locally instead of attacking the whole shape at once. This makes the simulations practical, fast, and accurate enough to use in real engineering.

Do you know how we simulate electromagnetic radiation around real systems like antennas, aircraft, and waveguides? We break the geometry into a mesh, a collection of small elements that lets the solver compute Maxwell’s equations locally instead of attacking the whole shape at once. This makes the simulations practical, fast, and accurate enough to use in real engineering.

Mathelirium

73,166 görüntüleme • 2 ay önce

When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes:. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the single line that unifies both viewpoints: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages!👌🏻 Once you see that, you can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematics

When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply the recipe, move on. Mechanical! Fuuuuck!😫😫😫😫 The problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. And that’s a big deal, because in real modeling the equations you meet are rarely nice enough to reward memorized recipes. So you end up trained to solve toy forms, while the actual subject...the behavior, the flow, the shape of solutions stays blurry. This is why I’m biased toward the old-timers. Their old-school way of doing things always surprises me:...they’ll spend time on one idea until it sticks, instead of sprinting through a syllabus checklist. One lecture from them and you start noticing a contrast. A lot of modern teaching feels like "finish the content,". You get marched through techniques, but you’re not left with a single thought that keeps bothering you later...the kind of thought that actually pushes you toward research-level curiosity. MIT OpenCourseWare’s Professor Arthur Mattuck did that to me in his very first ODE lecture. One lecture, and your whole relationship with dy/dx = f(x,y) changes. In this segment, Prof. Mattuck is basically saying: A first-order ODE is a slope field, and a solution is a curve that moves everywhere tangent to that field. The math breakdown Write the ODE as dy/dx = f(x,y). At each point (x,y) you attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes:. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the single line that unifies both viewpoints: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages!👌🏻 Once you see that, you can stop obsessing over whether you can write y(x) in closed form. You can start asking the questions that matter: where do solutions flow, where do they get trapped, where do they blow up, and where does existence/uniqueness fail just because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early and it’s exactly why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #MathAnimation #Mathematics

Mathelirium

53,338 görüntüleme • 5 ay önce

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 görüntüleme • 4 ay önce

For anyone to think they can deceive the international community in this digital age, you must either be out of touch or you believe everyone else is. The truth is simple: the world is watching, the world is informed, and you cannot hide reality anymore. The way out is to admit the truth, recognise the problem, and seek a real solution.

For anyone to think they can deceive the international community in this digital age, you must either be out of touch or you believe everyone else is. The truth is simple: the world is watching, the world is informed, and you cannot hide reality anymore. The way out is to admit the truth, recognise the problem, and seek a real solution.

DonAza

68,452 görüntüleme • 6 ay önce

“A visa is a visitor, it’s not a right … If you have the power to deny someone a visa before they get one, you most certainly have the power to revoke it once they get one and then do something they shouldn’t be doing.” – Secretary of State Marco Rubio

“A visa is a visitor, it’s not a right … If you have the power to deny someone a visa before they get one, you most certainly have the power to revoke it once they get one and then do something they shouldn’t be doing.” – Secretary of State Marco Rubio

U.S. Asia Pacific Media Hub

53,400 görüntüleme • 5 ay önce