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Thanks! "Tensors as matrices of matrices" isn't particularly deep; basically you can think of a 3D tensor of being a "cube" of entries, where each cross sectio

Thanks! "Tensors as matrices of matrices" isn't particularly deep; basically you can think of a 3D tensor of being a "cube" of entries, where each cross section in the direction going into the page is just a regular 2D matrix. For example, if you wanted to store the density of discrete voxels in a region of space, you could use such a tensor. This is in fact how MRI scan data is stored, as a bunch of cross sections that are glued together to make a 3D structure. You can then extend this to 4 dimensions, by taking an array of those 3D tensors, and so on. Understanding the link between wedge products and tensors, I think the key thing is the anti-symmetry of the wedge product, where if you reverse the order, the sign changes-- this is related to how the determinant is an "oriented" measure of area, and can take on negative values. The results of combining elements with the wedge product are called differential forms, and these can be thought of as "antisymmetric tensors." I agree that it's all a bit confusing. You can read more here: https://www.physicsforums.com/threads/how-do-tensor-and-wedge-products-relate-in-differential-geometry.150289/ https://www.physicsforums.com/threads/how-do-tensor-and-wedg... And ChatGPT can also do a good job explaing and answering questions about it. Hope that helped!

nsingh2 2y ago
I have a rudimentary understanding of differential geometry, but aren't tensors a bit more special than nd-arrays? I recall a tensor being an object that changes under coordinate transforms according to the jacobian of the transformation.
- eigenvalue 2y ago
  Yes, you're right, I was talking more about how to visualize a tensor. They also do have special transformation rules under coordinate changes where you have to compute the Jacobian to account for how the tensor "warps" regular Euclidean space locally.
082349872349872 2y ago
for antisymmetry via oriented measures, compare Stolfi, Primitives for Computational Geometry (1989): https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf
- vinnyvichy 2y ago
  Pluecker coords is the connection to Grassmann coords. (Aside: wheat in the heat, compare brevity in wit/spice in brevity) More gore less mess: Kidney in the Whitney
  - 082349872349872 2y ago
    aha, Plücker is how one gets Grassmann back down to dot/cross memory traffic? K is developer Whitney's, or Wit is threefold Halving, dedicatedly? I am disappointed there is no Spice Girls schlager/cover band named Die Würze in der Schürze. (although if I'm browsing YT for 90s covers, I guess that strongly implies my stage name should be "Old Spice") March: when the elite need to seed their wheat to beat the heat. (no more chill? get the drill!) EDIT: Sheaves, stalks, epis, germs, and fields — it just occurred to me that your noon demon in the rye might be metaphorical? (what is temperature in mathematics? the ratio of entropy delta, which makes sense, to energy delta, which requires a suitable def'n...)
    - vinnyvichy 2y ago
      And.. we circle back to Grothendieck.. Once you defn your equivalence classes, the partition fn, entropy, free energy etc is straightforward. Glad you caught on to the agronomic metagame. (Grassmann sadly a falsefriend, but agroSME maybe) Poem is close, but needs more oscillatory components (2D) -- heat to feed into elite need, eg. Microdose that homebrawn(/nighstilled) Roeteli into the gentle noon?
      - vinnyvichy 2y ago
        Aha Poire And also https://en.wikipedia.org/wiki/Grothendieck_inequality#Grothendieck_constant_of_order_d https://en.wikipedia.org/wiki/Grothendieck_inequality#Grothe...
        082349872349872 2y ago
        Seems highly likely that that J-L Krivine is (up to isomorphism, anyway) the same as the eponym of the https://en.wikipedia.org/wiki/Krivine_machine https://en.wikipedia.org/wiki/Krivine_machine . (being the advisor of J-Y Girard is evidence that « un train peut en cacher un autre ») EDIT: s/highly likely/certain/, see publication list: https://www.irif.fr/~krivine/ https://www.irif.fr/~krivine/

nsingh2 2y ago

I have a rudimentary understanding of differential geometry, but aren't tensors a bit more special than nd-arrays? I recall a tensor being an object that changes under coordinate transforms according to the jacobian of the transformation.

eigenvalue 2y ago
Yes, you're right, I was talking more about how to visualize a tensor. They also do have special transformation rules under coordinate changes where you have to compute the Jacobian to account for how the tensor "warps" regular Euclidean space locally.

082349872349872 2y ago

for antisymmetry via oriented measures, compare Stolfi, Primitives for Computational Geometry (1989): https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf https://bitsavers.org/pdf/dec/tech_reports/SRC-RR-36.pdf

vinnyvichy 2y ago
Pluecker coords is the connection to Grassmann coords. (Aside: wheat in the heat, compare brevity in wit/spice in brevity) More gore less mess: Kidney in the Whitney
- 082349872349872 2y ago
  aha, Plücker is how one gets Grassmann back down to dot/cross memory traffic? K is developer Whitney's, or Wit is threefold Halving, dedicatedly? I am disappointed there is no Spice Girls schlager/cover band named Die Würze in der Schürze. (although if I'm browsing YT for 90s covers, I guess that strongly implies my stage name should be "Old Spice") March: when the elite need to seed their wheat to beat the heat. (no more chill? get the drill!) EDIT: Sheaves, stalks, epis, germs, and fields — it just occurred to me that your noon demon in the rye might be metaphorical? (what is temperature in mathematics? the ratio of entropy delta, which makes sense, to energy delta, which requires a suitable def'n...)
  - vinnyvichy 2y ago
    And.. we circle back to Grothendieck.. Once you defn your equivalence classes, the partition fn, entropy, free energy etc is straightforward. Glad you caught on to the agronomic metagame. (Grassmann sadly a falsefriend, but agroSME maybe) Poem is close, but needs more oscillatory components (2D) -- heat to feed into elite need, eg. Microdose that homebrawn(/nighstilled) Roeteli into the gentle noon?
    - vinnyvichy 2y ago
      Aha Poire And also https://en.wikipedia.org/wiki/Grothendieck_inequality#Grothendieck_constant_of_order_d https://en.wikipedia.org/wiki/Grothendieck_inequality#Grothe...
      - 082349872349872 2y ago
        Seems highly likely that that J-L Krivine is (up to isomorphism, anyway) the same as the eponym of the https://en.wikipedia.org/wiki/Krivine_machine https://en.wikipedia.org/wiki/Krivine_machine . (being the advisor of J-Y Girard is evidence that « un train peut en cacher un autre ») EDIT: s/highly likely/certain/, see publication list: https://www.irif.fr/~krivine/ https://www.irif.fr/~krivine/