Gregor

@coexact.bsky.social

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only made of atoms

calathea orbifolia, a plant with a conical leaf
One of the referees asked: “Why not also a short comment at least on the effect of a surplus of 2, for instance?” Excellent idea! It turns out that if you toss a coin until the number of heads first exceeds the number of tails by +2 rather than +1, then the expected proportion of heads is ln 2 rather than π/4. Replacing +2 by a generic positive integer a appears to give rise to expressions involving π when a is odd and ln 2 when a is even, but we have not checked ChatGPT’s derivations and therefore regard its claims and formulas as conjectural. We leave further investigation to others
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Adams spectral sequence
It is a bottomless sea, but we do not have to worry about that. The picture of a bottomless sea is not so disturhing really . We just have to think of the situation near the surface,
One might instead be tempted to argue that the map V\otimes_{\mathbb{R}}V^*\to Hom(V,V) is surjective because it is an injective map between vector spaces of the same dimension (I’m not entirely sure how to prove it’s injective without picking a basis, but it might well be possible; however I do not know how to prove that the dimension of V\otimes W is the product of the dimensions of V and W without picking a basis :-/ ). Anyway, talking of dimensions, here is the other “basis-free” method I learnt to do these sorts of things: the canonical method to work out the determinant of an endomorphism without choosing a basis. If \phi:V\to V is a linear map, and if n is the dimension of V then \phi induces a linear map on top wedge powers \bigwedge^n(V)\to\bigwedge^n(V), and an endomorphism of a 1-dimensional space is canonically a number (proof: pick a basis :-/ and check it’s independent of the choice) which can be checked to be the determinant of \phi, and if you can do determinants then you can do char polys and hence traces.
We have arrived at the usual analytic behavior of a real time two-point correlation function in a thermal ensemble: two-point functions with different operator ordering are opposite boundary values of a single function that is holomorphic in a strip. We have found a strip of width 2πR, so the effective temperature is 1/2πR. A derivation that begins with the Euclidean correlation functions might be more transparent. Let tE = it be the Euclidean time. A Euclidean version of eqn. (5.28) is tE = R sin θ, x = R cos θ. (5.36) This is the thermal circle that is related to the observations of the accelerated observer. Let xE =(tE x ) . In Euclidean space, one considers the correlation function 〈O(xE (θ))O†(xE (0))〉. A priori, a Euclidean correlation function has no operator interpretation. To introduce an operator interpretation, one picks a direction as Euclidean time and introduces a transfer matrix that propagates operators in that direction. Then Euclidean correlation functions acquire an operator interpretation, with the operators being ordered in the direction of increasing Euclidean time. For example, if tE is chosen as the Euclidean time direction, then a general Euclidean two-point function is interpreted in the transfer matrix formalism as 〈O(tE , x)O†(t′ E , x′)〉 = { 〈Ω|O(tE , x)O†(t′ E , x′)|Ω〉 if tE ≥ t′ E 〈Ω|O†(t′ E , x′)O(tE , x)|Ω〉 if t′ E ≥ tE . (5.37) As before, this is consistent because if tE = t′ E , the operator ordering does not matter. Given this, the op- erator ordering in the operator interpretation of the Euclidean correlation function 〈O(xE (θ))O†(xE (0))〉 depends on the sign of tE = R sin θ, as in the above derivation. When we analytically continue 〈O(xE (θ))O†(xE (0))〉 from a function of θ to a function of τ = R(s + iθ), we get the two operator orderings depending on the sign of sin θ, as above. This distinction remains in the limit θ → 0±, where we recover the real time correlation functions with different operator orderings. 6 Algebras With a…
The Predictions I think life in 2025 would be really scary. Cars would be flying in the air. Planes will be flying on the ground. Cows will be flying int he air. There will be man-eating plants and man-eating beds. Man-eating footballs, pencils and baseballs. There would be aliens that would say MOO MOW. There will be cows that run around in circles like they are dumb. Cats will be 1 centimeter long. Cars will be flying on the ground. Houses will be haunted by ants and it will be scary. Andrew Baratucci Fourth Grade Paxtonia Elementary School
Google search result titled "D-modules and microlocal calculus" with description "This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus."
two reddit posts describing the same issue