Notes

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Software

Signal propagation - expansions

• Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives.
• How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)?

Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown - the change.

What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix?

An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces.

Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ.

Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed.

Study how turning the counting around relates to cycles - finite fields.

Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo?

• B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų

briauna = skirtingumas

Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is distinguished.

Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows.

One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow?

{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center.

Composition algebra. Doubling is related to duality.

Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them?

An relates to "center of mass". How does this relate to the asymmetry of whole and center?

Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė.

Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės?

Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas?

Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu?

Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution.

Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe.

Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix?

{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative.

Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu?

How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not.

Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą.

Bott periodicity is the basis for 8-fold folding and unfolding.

Use "this" and "that" as unmarked opposites - conjugates.

An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But:

• for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact.
• for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact.
• for Dn we have simply "this" and "that" (not-this).

Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out.

Duality examples (conjugates)

• complex number "i" is not one number - it is a pair of numbers that are the square roots of -1
• spinors likewise
• Dn where n=2
• the smallest cross-polytope with 2 vertices
• taking a sphere and identifying antipodal elements - this is a famous group
• polar conjugates in projective geometry (see Wildberger)

Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page.

If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root.

This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between.

Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement.

Massless particles acquire mass through symmetry breaking: Yang-Mills theory.

Geometric unity I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better.

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