How to get URL link on X (Twitter) App
https://twitter.com/vikhyatk/status/1969151806501487096The tensor product ⊗ is, conceptually, the most general (binary) operation that behaves "how a product should behave". In practice, this means that the order of brackets shouldn't matter, i.e. X⊗(Y⊗Z) should be the same as (X⊗Y)⊗Z, for any objects X, Y and Z... (2/9)
In this intuitive picture, the two "holes" of the straw are 1-dimensional circles, and they're connected by a 2-dimensional cylinder (the straw itself). Mathematically, this relationship is called a "cobordism". Two n-dimensional manifolds are "cobordant" if they form... (2/20)
https://twitter.com/EliotJacobson/status/1942937846484177239"First, I tried <standard method>, but it failed for <enlightening reason>, so I investigated whether I could exploit this fact to find <counterexample> with <property>, but all objects obtained through this technique ended up having <interesting property> in common.... (2/4)
Rather, everything travels through both space *and* time, simultaneously, with a speed of c. If you're standing still, then all of your velocity is focused in the time direction (with none in the space directions), so you move through time with a speed of c. (2/4)
https://twitter.com/ThePhysicsMemes/status/1927662467481469428

The "net flux” of momentum vectors through the surface is zero, since every momentum vector poking through the surface in one direction is counteracted by a momentum vector poking through in the opposite direction. In other words, the disk has no *linear* momentum. (2/14)

Link: https://twitter.com/getjonwithit/status/1911933388895756371When attempting to model the world computationally, there are typically *three* computations that one needs to consider: the computation that the system itself (e.g. the universe) performs, the computation that the observer performs, and the "encoding function". (2/14)

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When you first encounter the sine function, you probably see it defined in terms of triangles, which means that it really only makes sense for x between -pi and pi (or -180 and 180 degrees). But later on, you learn that the "true" sine function is much larger. (2/17)
Suppose that the room you're in defines a 3-dimensional coordinate system (x, y, z). If the walls in the x-direction get pulled apart in the x-direction, and those in the y-direction get pulled apart in the y-direction at the same rate, etc., then the room is expanding. (2/12)
https://twitter.com/getjonwithit/status/1878094729809723568

Here's one way to think about Birkhoff's theorem: suppose that you have a perfectly spherically-symmetric object that is uncharged and not rotating. Clearly, by measuring its gravitational field (i.e. the extent to which it curves spacetime), you can determine its mass. (2/14)
First, it is crucial to distinguish two things: a computation (which we can formalize as the operation of a Turing machine), and a simulation (which is a specific type of computation, whose output states are interpreted as approximate states of some *external* system). (2/10)

When you throw an object into the air and try to calculate when it will hit the ground, you can do this (neglecting air resistance) using a SUVAT equation: by solving for t in s = ut + 1/2 at^2. But this equation is a quadratic, and so generically has 2 solutions. (2/14)

A "knot" is an embedding of a circle (S^1) in 3D space (R^3). A "link" is a collection of knots (potentially linked together, but not intersecting). And whenever you smoothly (topologically) deform a knot or link, some properties will always remain the same. (2/16)

In previous work (), I showed that black holes in discrete spacetime accrete matter more slowly than their continuous spacetime counterparts, for essentially ergodic reasons: with only countable degrees of freedom,... (2/6)arxiv.org/abs/2402.02331
Suppose that you have a big collection of matrices, representing elements of some symmetry group. For instance, the elements of the 2D rotation groups U(1) or SO(2) can be represented by 2x2 rotation matrices. Call this collection a "representation" of the group. (2/17)

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As soon as we start extending sine and cosine to the complex plane, we see that they start blowing up - they don't remain nicely bounded like they do over the reals. We could try imposing bounds on them, but that would quickly break their differentiability. (2/9)