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While revising group theory for a graduate abstract algebra course, I went looking for a second voice, because algebra has this specific kind of pain…you can follow every line of a proof and still lose the idea. That’s how I landed on Benedict H. Gross’s Harvard Extension Math 122...

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Mathelirium

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34,256 Aufrufe • vor 7 Monaten •via X (Twitter)

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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
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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

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Paul Azinger takes a shot at Rory McIlroy’s Ryder Cup behaviour: “Today in the press conference he said ‘I think golf should be held to a higher standard of decorum. But in the meantime he says ‘f-you, f-you, f-you’ in full voice for the world to see. “He turns around and says to the guy ‘shut the f up’. The guy in the media today asks him ‘how did it feel Rory to tell the guy to shut the f up and then hit it to feet’ and he said ‘it felt pretty f-ing good.’ “And I’m like, which is it Rory? Is it that golf is held to a higher standard or are you just gonna f you the fans and act like that’s ok?” What are your thoughts on this? Is Azinger right? Or was Rory justified in his actions? Clip from Subpar with Colt Knost and Drew Stoltz full link in comments
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Paul Azinger takes a shot at Rory McIlroy’s Ryder Cup behaviour: “Today in the press conference he said ‘I think golf should be held to a higher standard of decorum. But in the meantime he says ‘f-you, f-you, f-you’ in full voice for the world to see. “He turns around and says to the guy ‘shut the f up’. The guy in the media today asks him ‘how did it feel Rory to tell the guy to shut the f up and then hit it to feet’ and he said ‘it felt pretty f-ing good.’ “And I’m like, which is it Rory? Is it that golf is held to a higher standard or are you just gonna f you the fans and act like that’s ok?” What are your thoughts on this? Is Azinger right? Or was Rory justified in his actions? Clip from Subpar with Colt Knost and Drew Stoltz full link in comments

Flushing It

1,056,153 Aufrufe • vor 11 Monaten

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
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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

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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
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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

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The woman who went viral for calling a man ‘Not Real’ on an American Airlines plane has been identified as Tiffany Gomas. She claimed: “I’m telling you, I'm getting the f*** off and there's a reason why I'm getting the f*** off and everyone can either believe it or they can not believe it,' the unidentified woman is heard saying as she walks to the front of the plane. I don’t give two f****, but I am telling you right now - that m*****f***** back there is NOT real. And you can sit on this plane and you can die with them or not. I'm not going to.” Why is America so obsessed with this woman who was obviously having a really bad day?
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The woman who went viral for calling a man ‘Not Real’ on an American Airlines plane has been identified as Tiffany Gomas. She claimed: “I’m telling you, I'm getting the f*** off and there's a reason why I'm getting the f*** off and everyone can either believe it or they can not believe it,' the unidentified woman is heard saying as she walks to the front of the plane. I don’t give two f****, but I am telling you right now - that m*****f***** back there is NOT real. And you can sit on this plane and you can die with them or not. I'm not going to.” Why is America so obsessed with this woman who was obviously having a really bad day?

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Fagnani: Do you still attend Sociology? Zeudi: Yes, I still attend—I’m still enrolled. F: Because when people ask you how long you have left until graduation, or when you’ll graduate, you don’t take it very well. In a livestream you said: “I’ll do university whenever the fuck I want. You’re a pain in the ass and make me anxious. I’ll graduate when I feel like it. Right now I’m doing something else. I attend university for personal culture, not to say: I have a degree, I have a piece of paper. I don’t give a damn. You’re a pain in the ass, my God.” You took it very well, I see? Zeudi: I’ll explain why. I think people shouldn’t put too much pressure on young people who are enrolled in university. Maybe I’ll also send a message to parents. But it’s true, because there are people who feel the pressure of university and can’t manage to pass their exams. F: So let’s make an appeal: study calmly, don’t rush. Take 15 years—it’s the same. Z: Yes, it’s the same, 15 years—it’s the same. F: Imagine I wanted to ask when you’ll graduate, I’m scared. Z: I don’t know when I’ll graduate, honestly. F: I’m joking, it was a joke. Z: Still, the fact is that everyone has their own timing. I think that’s the message that should come across. F: Let’s say that if a degree program is meant to take four years, there’s a reason. Z: Of course, that’s true. But I didn’t choose Sociology. F: Was it something forced on you? Z: Yes, because I liked the idea of continuing to study.
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Fagnani: Do you still attend Sociology? Zeudi: Yes, I still attend—I’m still enrolled. F: Because when people ask you how long you have left until graduation, or when you’ll graduate, you don’t take it very well. In a livestream you said: “I’ll do university whenever the fuck I want. You’re a pain in the ass and make me anxious. I’ll graduate when I feel like it. Right now I’m doing something else. I attend university for personal culture, not to say: I have a degree, I have a piece of paper. I don’t give a damn. You’re a pain in the ass, my God.” You took it very well, I see? Zeudi: I’ll explain why. I think people shouldn’t put too much pressure on young people who are enrolled in university. Maybe I’ll also send a message to parents. But it’s true, because there are people who feel the pressure of university and can’t manage to pass their exams. F: So let’s make an appeal: study calmly, don’t rush. Take 15 years—it’s the same. Z: Yes, it’s the same, 15 years—it’s the same. F: Imagine I wanted to ask when you’ll graduate, I’m scared. Z: I don’t know when I’ll graduate, honestly. F: I’m joking, it was a joke. Z: Still, the fact is that everyone has their own timing. I think that’s the message that should come across. F: Let’s say that if a degree program is meant to take four years, there’s a reason. Z: Of course, that’s true. But I didn’t choose Sociology. F: Was it something forced on you? Z: Yes, because I liked the idea of continuing to study.

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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
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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

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Just in: “VeryDarkMan has over 40 Court Cases he is not above the law, he is lucky he was not Picked up by DSS, he is sending a wrong message to the youths that’s it’s ok to disobey the law and nothing will happen to you. VDM has over 4million followers and posts carelessly with sensitive issues that can throw nigeria into a state of anarchy, if you want to f!ght for the people f!ght constructively” - Nigerian lady !!! 🚴🏿‍♀️🚴🏿‍♀️🚴🏿‍♀️💨
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Just in: “VeryDarkMan has over 40 Court Cases he is not above the law, he is lucky he was not Picked up by DSS, he is sending a wrong message to the youths that’s it’s ok to disobey the law and nothing will happen to you. VDM has over 4million followers and posts carelessly with sensitive issues that can throw nigeria into a state of anarchy, if you want to f!ght for the people f!ght constructively” - Nigerian lady !!! 🚴🏿‍♀️🚴🏿‍♀️🚴🏿‍♀️💨

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🚨NEW: Scott Adams *GOES SCORCHED EARTH* on James Carville over clip I made of him calling for Trump "collaborators" to be shaved, marched and spat on after he leaves office🚨 "As long as this f*cking a**h*ole is talking like this, I’m in favor of a third term. If you can calm down this bullsh*t, I’ll be on your side — no third terms for anybody." "That’s where I was 10 minutes ago. Now I'm looking at this and I’m saying, 'F*ck you. F*ck you.'" "If you’re going to give this vague threat that when Trump leaves, people like me are going to be victimized — yeah, we're going to go for a third term. And that's on you a**h*le. That’s all on you, a**h*le." "Because I was absolutely 100% against a third term until right f*cking now. And the longer I look at this, the madder I get." "Yeah — third term. Or shut the f*ck up about this 'collaborator getting spit on' sh*t. You need to take that back right away." "And until he takes it back, I’m all down on a third term."
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🚨NEW: Scott Adams *GOES SCORCHED EARTH* on James Carville over clip I made of him calling for Trump "collaborators" to be shaved, marched and spat on after he leaves office🚨 "As long as this f*cking a**h*ole is talking like this, I’m in favor of a third term. If you can calm down this bullsh*t, I’ll be on your side — no third terms for anybody." "That’s where I was 10 minutes ago. Now I'm looking at this and I’m saying, 'F*ck you. F*ck you.'" "If you’re going to give this vague threat that when Trump leaves, people like me are going to be victimized — yeah, we're going to go for a third term. And that's on you a**h*le. That’s all on you, a**h*le." "Because I was absolutely 100% against a third term until right f*cking now. And the longer I look at this, the madder I get." "Yeah — third term. Or shut the f*ck up about this 'collaborator getting spit on' sh*t. You need to take that back right away." "And until he takes it back, I’m all down on a third term."

Jason Cohen 🇺🇸

846,417 Aufrufe • vor 10 Monaten

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Sean Strickland speaks about how BRUTAL the pay structure is in the UFC, going as far as to call the pay structure PREDATORY 😳 “When you’re a fighter you’re so poor… 20 grand for a fight, so you pay off your manager, you pay off your taxes, and if you lose you really only get like 8-7 grand…” “The UFC is f-cking brutal, the UFC is so predatory. This is why you see a lot of foreigners excel, because guys in Brazil that live in the f-cking favelas, 10 grand to them is f-cking life changing… You can’t even f-cking pay rent for a couple months here…” (Via Shawn Ryan Show)

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RedWave Press

20,008 Aufrufe • vor 3 Monaten

Fagnani: Tell me a bold move you made that you really own. Zeudi: I won Miss Italy and I said no to Valeria Golino. F: Oh… amazing. Z: You know, right? At the time I was stressed and couldn’t understand what was happening to me because of so many situations. F: For a movie she called you? Z: No, for a casting, an audition, where Golino was present. F: Of course. Z: “The Art of Joy.” I was supposed to audition for the lead role. F: And you didn’t show up. Z: No, I said no to Golino. That’s a bold move. F: But that was against yourself, though. Z: Yes, unconsciously. Later I understood. F: Did you know who Golino was when she called you? Z: At first, I didn’t. The point is, this was a bold move because I kept saying no, no, no. The third time, I went to the audition. F; Ah, you went? Z: Of course. I remember Valeria kissed my hand and said, “Thank you.” F: “Thank you” is fine. Z: No, before the audition, like, “Thank you for coming.” I was crazy. Now that I think about it, I say: “Madness.” She probably doesn’t even remember. But madness, because it was a huge opportunity I lost due to my unawareness and immaturity. F: And how was it, were you good at the audition? Z: I wasn’t good because I learned the script in the dressing room right before going in. F: You hadn’t studied. Z: I hadn’t studied. F: And that’s the bold move you made against yourself. 😭😭 #ZeudixBelve
1:34

Fagnani: Tell me a bold move you made that you really own. Zeudi: I won Miss Italy and I said no to Valeria Golino. F: Oh… amazing. Z: You know, right? At the time I was stressed and couldn’t understand what was happening to me because of so many situations. F: For a movie she called you? Z: No, for a casting, an audition, where Golino was present. F: Of course. Z: “The Art of Joy.” I was supposed to audition for the lead role. F: And you didn’t show up. Z: No, I said no to Golino. That’s a bold move. F: But that was against yourself, though. Z: Yes, unconsciously. Later I understood. F: Did you know who Golino was when she called you? Z: At first, I didn’t. The point is, this was a bold move because I kept saying no, no, no. The third time, I went to the audition. F; Ah, you went? Z: Of course. I remember Valeria kissed my hand and said, “Thank you.” F: “Thank you” is fine. Z: No, before the audition, like, “Thank you for coming.” I was crazy. Now that I think about it, I say: “Madness.” She probably doesn’t even remember. But madness, because it was a huge opportunity I lost due to my unawareness and immaturity. F: And how was it, were you good at the audition? Z: I wasn’t good because I learned the script in the dressing room right before going in. F: You hadn’t studied. Z: I hadn’t studied. F: And that’s the bold move you made against yourself. 😭😭 #ZeudixBelve

Korslayage

18,940 Aufrufe • vor 4 Monaten

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 #
20:13

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

40,841 Aufrufe • vor 6 Monaten

🚨 WOW. Somali man intimidates and harasses Nick Shirley in Minneapolis so much security had to step in. He is HEATED. "Are you f*cking mad?! Are you STUPID?!" *Security pushing him off* "What if a MASS SH00TING happens because of you? If ANY Somali gets hurt, it's because of NICK SHIRLEY!" "You son of a b**ch! Get the F out!" SHIRLEY: Can you tell me where the transportation company is? MAN: "Get the F out of our community, you f***ing idiot!" *Cars start honking* It's like an ambush. Nick shirley is being targeted by the Somalis. Stay safe.
1:42

🚨 WOW. Somali man intimidates and harasses Nick Shirley in Minneapolis so much security had to step in. He is HEATED. "Are you f*cking mad?! Are you STUPID?!" *Security pushing him off* "What if a MASS SH00TING happens because of you? If ANY Somali gets hurt, it's because of NICK SHIRLEY!" "You son of a b**ch! Get the F out!" SHIRLEY: Can you tell me where the transportation company is? MAN: "Get the F out of our community, you f***ing idiot!" *Cars start honking* It's like an ambush. Nick shirley is being targeted by the Somalis. Stay safe.

Eric Daugherty

86,642 Aufrufe • vor 7 Monaten

Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f*(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f*(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering
0:50

Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f*(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f*(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering

Mathelirium

38,506 Aufrufe • vor 8 Monaten

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
0:46

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 Aufrufe • vor 6 Monaten

Joe Rogan GOES OFF on Democrats’ “tax the rich” slogan 😂 “F*ck you!” “The thing that drives me the most nuts is when these progressive talking heads say, ‘They don’t want to pay their fair share.’” “With the amount of waste and fraud, why would you?” “You don’t think there should be some accountability to how much f*cking waste and fraud that has been clearly demonstrated? Like, the solution is just give more money?” “Where do you think this money is going to go? Where it’s actually going to help people and affect things in a positive way? There’s been no indication that that’s the case—that the real problem is they just haven’t had enough money from the billionaires. That’s f*cking ludicrous. That idea is ludicrous.” “It’s such a lazy, intellectually lazy way of framing this whole discussion that saying, ‘oh, they don’t want to pay their fair share.’ F*ck you!”
1:07

Joe Rogan GOES OFF on Democrats’ “tax the rich” slogan 😂 “F*ck you!” “The thing that drives me the most nuts is when these progressive talking heads say, ‘They don’t want to pay their fair share.’” “With the amount of waste and fraud, why would you?” “You don’t think there should be some accountability to how much f*cking waste and fraud that has been clearly demonstrated? Like, the solution is just give more money?” “Where do you think this money is going to go? Where it’s actually going to help people and affect things in a positive way? There’s been no indication that that’s the case—that the real problem is they just haven’t had enough money from the billionaires. That’s f*cking ludicrous. That idea is ludicrous.” “It’s such a lazy, intellectually lazy way of framing this whole discussion that saying, ‘oh, they don’t want to pay their fair share.’ F*ck you!”

The Vigilant Fox 🦊

298,439 Aufrufe • vor 5 Monaten