I remember being lured at the time by Irrlicht learning curve for a professional project and migrating to Ogre in the middle... How nightmarish was using assets coming from 3D authoring software. Repressed memories return about switching from 3D studio to XSI (better) and fighting with Collada converter plug-ins quality issues. Prefer the dentist.
Section #7 is on the Langlands Conjectures. "Conjetures in a proof blueprint?" Curious about the role in the proof.
"The { Wiles } modularity theorem { that cracked FLT } is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. (https://tinyurl.com/yc6kth2m)"
New math is emerging "as we speak" from the group of Gaitsgory (https://tinyurl.com/9r5bsufj), but apply to the (differential) geometric Langlands area, while the quote looks about the algebraic-geometry area (of Weil's triple Rosetta Stone). Yet areas have well understood connections. Both kind of works (new math and formalizing) are very needed. Verifiable proofs have to help to avoid math building collapsing under own weight/richness.
This has the perfect qoutation for the well known fact of the Galileo->Lorentz group overthrow from an indisputable source:
"The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. The restricted theory { 1905 } changed our ideas of space and time in a way that may be summarised by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. The latter group is a much more beautiful thing than the former - in fact, the former would be called mathematically a degenerate special case of the latter { c->∞ }. The general theory of relativity { 1915 } involved another step of a rather similar character" { diffeomorphism group/category }. I came to think lately that much of the basic groups in physics, Lorentz and gauge, have all more or less rotatory features.
I think the rotations in the Lorentz group just reflect the isotropy of space, which comes down to the quite natural idea that if you observe a physical system from a different direction it doesn't change its dynamics. The full symmetry group of special relativity is the Poincare group, which includes the spatial translations, reflecting homogeneity of space.
The gauge groups are interesting in being extra symmetries beyond the spacetime ones, and yet they're closely related to spacetime symmetries, e.g. SU(2) being the double cover of the rotations SO(3). I also find it interesting that the groups that are physically basic, such as SU(2) being the one required to represent the phenomenon of spin, are also mathematically significant, in this case since SU(2) is the unique simply-connected group associated with the shared Lie algebra of SU(2) and SO(3). That shows some kind of deep connection between mathematics and physics. I'm just at the beginning stages of learning QFT and differential geometry so I don't have a feel for why that is or what it means at an intuitive level, and haven't seen any explanation for it. I think at the moment it's just a feature of our deepest experimentally-verified theory and so it would need a deeper theory to explain it.
Things are kinda weird. We had a reasonable notion of "now", as in... a photo. Came 1905 relat. and that blows the idea up, there is no more a consensual concept of "now". But though not consensual, the photo definition, even if dependent on the observer, is still there. One thing that an earlier documentary on the Webb experiment impressed on me is that with that "now" so defined, the farther the things I see are the earlier, so it is legit that we are seeing big bang "now" (modulo some shit occluding). It is geometrically also weird. We can look around 2D in 360 degrees and see bbang photons coming from everywhere, look 3D spherically around every direction receiving bbang photons. Yet all photons are sourced in a single point-like event (bbang, or post bbang "first light" in fortunate ESA terms). Photons came to us from behind in spacetime from our causal past light-cone, but that cone joins base with the forward oriented bbang causal future light cone, so any valid signal one gets must be inside that region of joined cones. The pic has to be the same as in "suspension" in topology wiki page.
The finite speed of light was known before 1905. It was known by 1676.
Second, we see "big bang photons" in the CMB, but we're not seeing the big bang. We're seeing the glow from stuff at the time when the universe became transparent, that is, no longer a plasma.
Declarative representations are also very groovy, I loved the HarmTrace guys [1] view of harmonic analysis as parsing by a grammar so the AST reflects the harmony of the piece (for instance secondary dominants are similar to subordinate clauses in normal language). It is incomplete modeling sure, yet it generates a infinite variety from a finite set of generators, that make sense tonally (hand picked, not machine-inferred).
If all you mind is the EE curriculum then ok. Or else there is an interesting work of Gerald Westendorp on the web [1] on how allowing other classical passive components (Ls & Cs) you can get discretizations (and hence alternative views) of a very wide class of iconic Physics partial differential equations (to the point that the question is more what cannot be fit to this technique). G. W. is alive and kicking in mathstodon.
But this is not using bitwise AND, just the Pascal's triangle approach. (Interestingly, you can reformulate that as a neighborhood-2 2-state 1-dimensional cellular automaton pretty easily; it occurs in a couple of different guises in Wolfram's catalog.)
Here's an ASCII-art version that uses AND as Michał describes:
32 value size : line cr size 0 do dup i and if bl else [char] # then dup emit emit loop drop ;
: pasand size 0 do i line loop ;
That looks nicer than my version. But you should put the `cr` before the inner loop, not after it. That way you can remove the `cr` before the outer loop.
Nothing much to do with your great post, but I almost REALLY liked that first pyramid, but the last line being off threw me visually, so I had to straighten it out:
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