How can we map the Heisenberg group onto a flat linear space?

[deferro]

A cylinder is aligned along zz' and both ends are closed, one with a rigid plate which has a smaller tube and valve attached.

If the other end is covered with an elastic material and the edges are thin enough so that at time t, if the pressure is lowered in the cylinder by evacuating air, the sheet breaks and begins to travel toward z' calculate the image at time t' if the elastic constant is k. Assume the image has a radius of one unit of distance and the frictional force between edges is linear. Derive a general form for a sheet with a differential thickness of material including sheets with thick but infinitesimally small radii, with a puncture at the center p which draws air from z to z' through a small channel one molecule of air wide; assume a small amount of energy is lost for each molecule transported into the near vacuum, which is recovered as heat in the material.

Each molecule carries vibrational and rotational momentum into the void at constant T, or ambient background.

Do I need a triple, like a single and a duple here? If S the elastic surface isn't punctured it will deform linearly which depends on the material's geometry, how isotropic for the initial uniform case, so can use a straightforward deformation since diffusion is constant and so is friction, the sheet will be trying to restore its shape. Then the pressure is the deformation operator and the image is a sheet curved or expanded into 3 from 2 dimensions, all the curvature is written by the forces experienced be cause of kP, if dT is also internally conserved, the temperature of the cylindrical system should stay ambient over sufficient periods for the defomation, since evacuation can be brought to the point of separation of the circular sheet from the edges, which is the 'launch', a volume measure, that's where the algebra is Can the generalization be viewed as a kind of signal loss per molecule for the punctured sheet?

Anyone have pointers here, please? I've read something about a fluid in a long flattened tube and pressure images and want to know more. The image is a volume product or form, isn't it?

 

[deferro]

Actually I think I do know a little more at least about calculating the initial state. At minus launch assume ambient Td for deformation due to heat transfer, then at launch the sheet is deformed so the total space inside the cylinder is the set of points tangent to the inner surface.

This has a volume measure V(0) for t(launch) and this is the product that needs pressure transformed into a volume form w, so that V,w -> V(t') . It has to relate the tangent bundle Tc to the deformation in the sheet at V(0). Linear forces are conserved and friction is assumed linear at the shear point. Pressure can be assumed to stay constant at t(launch until t'.

Ok so the total space is then the initial bundle on the form, and it's imaged by the sheet, which was in S, and is now in S', traveling along a pressure wave of momentum restricted by the sides which are also S, s.t. S' - S is the volume change, due to V,w'

So it's Fp the pressure - (Fl the linear travel + FD the deformation) = I the image;
All forces are generated by a density/pressure Hamiltonian H, the initial or rather trivial bundle is the circle on the sheet at prelaunch; then H(t) is the volume operator at launch V,w -> t,t' and we need the dynamic functional H() -> H(0) + H(t), s.t. V,w(t) -> V,w(t').

 

[deferro]

My grasp of Dirac notation is a little rudimentary, but i understand it's a topographic representation of an abstract edge-vertex or horizon-surface type model, or just a place to have things that you can transpose from one to the other, so that the Dirac graph G(V,E) is the notation g('<,>','|'), in which g'('|',{'>','<'}) transposes edges with vertices. as g(a)(<,>,|) -> <a>; <a|b>; <a|; |a>, and for which sequence g'(a) -> |a|; |a><b|; |a>; <a|.
So it just says a is against (a surface or at a vertex), it has an edge if it is positioned on the surface (or at a vertex), it has two edges if it isn't. The edges point left/up and right/down. IOW, if it has no edges, there is no single location on a surface because it's against or between two, which state can also be a vertex, so that the vertex-edge meaning changes readily.
Then p for the linear momentum is generated by H -> H(t,t'), where H is initial ambient and the deformation is zero; pressure is the volume measure V,w so H(w,t) -> H(ambient) + V,w(t,t')

and we're away laughing.

 

[deferro]

Next, we need the B or bundle in the Hamiltonian, and the measure operator that keeps B on the transported sheet, conventionally uB; this is exponentiated by the algebra of the state, or need the state eqn for an ambient system at Tp and pressure P; air is fairly well-understood so that's taken care of.

Pressure-density in a gas is because of its temperature and molecular geometry = a rotation group w/polynomials in t -> check diophantine approximations. Need convex hull or partition of P(V,E) and T(V,E) for G|Tp , and subgroups in the base space B - (total space).

 

[deferro]

And the map B, is of course the Bode-approximation that is the pole-zero response; the convex hull is partitioned by |H| proportionally to 1/w, the volume operator. So far so easy.

We need a pole and zero for a linear impedance, Z and an elastic sheet S against a cylinder Sc, with <w|(Tp.k).

the formulation is a complex response in t, with the frequency s the sigma fn, written H(s) = k/(s - p'), p' real-valued in t. So that H(s) has two components in a phasor p.
An Argand rep is the d-approximation for G, w/ ln(p) dimensions in t.
|H| is asymptotic in s, and at low w is linear or k/w'; at high w its -p' or the system is absorbing maximum pv from the density operator, in the system. Then v = p'V is the forces recovering energy in s, over t.

 

[deferro]

Here endeth the lesson

We have bundle map B, as <u|H|v> taking the image I|G(u,v) s.t. I(t) -> |I(v) -I(u)| over G the system,
Tc is the cylinder bundle on Tp,Tv the bundles in B|(p,v) -> total response, and S,S', with Z-images I(c,p,v) and an elastic k, when Tc is assumed constant or the cylinder has no (system) response.

There are two functions that map the partitions in S|S' to the image. You need a fork f, to hold the solid S, so you can section or divide (cut) S with k, the always (constantly) sharp knife; f has to be continuous over G and the 'constant' function has to slice or partition the space appropriately -> proper function in time.

The rubric here is, "to use a knife k to cut Sn, S(p,q), Si, S(m,{x2n,1}), ... you need fork f so u can k m to H(m), and measure the inner 'cost' for the partition, please pay p at counter Tc"

We need a sectional f to pin linear momentum, or L; we have the bundle maps (E,B,F,S), and need F -> E -> S -> B, we have the preimage I|S which is 2pi, and is the trivial bundle, (B,E)|V is a phase space with stochastic map (subalgebra) B -> uB.

And with a general version (this is a mere outline of tangent bundles over a phase space), I can get to a bipartite graph G (I have this in a toric code in fact) with an abstract gauge g, which writes w on T a tangent bundle.

L|F extends w(o,1) -> [0,1] for observer o, at (x,y,z) = (0,0,0) or null-vertex. F -> f(0,1) is conserved as the continuous map F|[0,1] f in S1.

c(t) is a curve, the path written over T, against its surface impedance FD the sum of deformations in the image I|Tc; f(p) is a probability or stochastic measure L'|C(m), for a vertex m in a "spin network" graph G(s,t).

So far I've mapped it to: $$A\, =\, \frac { (5 \sqrt 3)4.\fractur{\hat r{^3}_{(u)}}) } {\tau \sqrt 5}$$ A complex phase space that I'm exploring with it. It's Godelian/Eulerian I hope. with a (w,m) function and a density of states uB -> B|p(m)

 

[deferro]

Actually A can be more or less any real or imaginary "area" with extra dimensions I suppose.
I'm analyzing a Coxeter group that is a prime generator in the sense a volume is created with numbers (permutations) in it. This has a natural gauge, n a counting number - and you can count backwards too.

However, after n counts that are permuted by a sectioning function, (f,q) the f and q components diverge over G the graph (Hamiltonian cycles). There is an involution that resolves both prime factors in each subgroup Pn of primes and a generator. There's a volume V, and again a spectral theory approach applies. The sheet model is geometric, with a punctured sheet, traveling or at the breakpoint,

There's another flow q, for the Tp bundle on B; q is the subgroup of rotational + vibrational momentum for p the linear flow of molecules of air, so in that sense the puncture permutes air molecules from the group N,O,H20, ...; of course a N atmosphere or other monomolecular gas applies, or, the group (of molecules in air) can be restricted, say to 3 kinds or whatever. Or it could be a Penning mix(ed state) of inert gases, the sheet could be made of a glassy or quasicrystal material, etc.

Anyhoo:

B-map:
fiber is trefoil over the cube in R3.

n 	f	q
0 	1 	1
1 	9 	6
2 	54 	27
3 	321 	120
4 	1847 	534
5 	9992 	2256
6 	50136 	8969
7 	227536 	33058
8 	870072 	114149
9 	1887748 360508
10 	623800 	930588
11 	2644 	1350852
12 		782536
13 		90280
14 		276

Generator is of permutations in groups, function is U(1)x(3)f; generates subgroup of prime factors in each group.
Section is f(n) for rotations of pi; for q(n) rotations in the group sections are subgroups of f' for n the index of permutation group spaces.
Sectioning is f' o q for n, q o f for subgroup generators.

Crossings are in the Mobius rotation group S'xS' on the trefoil so a volume measure is available with a transform.

The V is expanded nullspace {} -> {n}|(q o f, f' o q); initial H is f0,q0 and n goes 0 -> 1 for/at trivial uB;
Deformation over the cube/polytope is then a measure on the Coxeter group in G(V,P,H) which has a triple in S'xS'; phase of f,q over the space is in map B above

Color index is i,j over a face so that, any abstract function is assignable or encodable in primes or partial primes in P(p,q).

u is rotations in U(1) -> A2, etc.

Recurrence:
F(r), Q(r') for full, quarter turns = pi, pi/2 rotations r,r' -> R(r,r')
Rr = (e,e') -> (e',e''); R'r = (e'',e') -> (e',e)
nR(r,r') = Rr(e,e') -> (e',e''); nR'(r,r') = Rr(e',e) -> (e'',e')

F(r): initial state {0} is no colors

0xF(r,r') ->  (1)p +-0 = 1T, {3x1/3 + (0)} 
; (3)f faces, 1 color map (in ground state p(0,0))
f1.  (9+f(0))p = 10T (5x2),  f0 + 8 + 1    {f0 + 3x3 + (0)} 
; (3)fx3 + initial break p(0,1)
f2.   (54+f(1))p = 64T (8x8), f1 + 45 + .. {f1 + (3x3x3)fx2 + (0)}
; p(0,2) section of perm bundle is at launch state for prime factors
f3.   (321+f(2)) = 385T, (5x7x11), f2 + 267 + ... {f2 + (3)fx89' + ...} 
; p(1,3) f3 is prime generator p2+0,3 
f4    (1847+f(3)) = 2232T, f3 + 1526 + ...  {f3 + (3)fx109' + ...}           
 ; p(2,4) p3+1,4 " 
f5    (9992+f(4)) = 12224T, f4 + 8145 + ... {f4 + (2x2x2)fx1249' + ...}   
 ; p(3,5) p4+1,5
f6    (50136+f(5)) = 62360T; f5 + ...    
; p(4,6)

... is where I'm up to (in a rough way) with the Coxeter Pn set in A2, as S1xS1 crossings on the trefoil, or as crossings on a flattened Mobius strip, so that the edges describe an internal empty space (V0) with 3 edges and can be inscribed so the circle touches the vertices of the triangle (inside the loop).

Does anyone have any comments? (about the notes, or, what isn't not(confusing)?). I've extended this model to a simple bipartite graph which describes the phase space of a spinning coin - quantum coins are what you might call fair and balanced if you balance them and so, you can cheat with them--this is quite handy, in terms of saving time.

And I usually would get hauled up by experts insisting that describing the following indicates a confused mind (so here we go at this venue):

Assume a scalar gauge g, with a force Fg, so that total forces in a frame I,C with Cartesian coordinates, I fixed the other traveling along the x direction, is F(t) = Fg + FL + FD +FC; L is linear, FD is the result of deformations in I|C, and FC is all remaining forces, which are initially assumed negligible. Sounds like the US army ballistic engineer's manual (wait asec, it was ..doh!.)

P.S. Because the pocket cube is reversible from any subgroup - the chart shows that F(0) is always 11 f rotations or no more than 14 q ones from the ground state, but successive primes as factors of the 'trivial bundle' (2,3,5,7,11) fall out starting at f3' = 89. Reversibility means the primes are affine products of Clifford algebras where union over joint and disjoint spaces lifts a flattened Mobius loop from the plane, to the bundle on the cube. The 2-slice group is 3+3 colored and each color evolves imaginary products, since AND normally means conjoint union - you would need to glue faces to each other . It represents a 'free logic' in that it's Hamiltonians are, the cube is always 'decolored in nullspace' since you can peel them off.
By F(0), I mean the projected face(s), which is (1,2,3); this can be "any" pattern, but is constrained to 6 for the pc color-prime generator (in s,t, natch).

The elastic sheet with puncture measures individual particles of air in a Laplacian way; the H20 molecule is a graph with degrees of freedom and approximations here are in the domain of geometry of small inertial bodies with oscillating pendular motion (rotations and vibrations) and a path which is statistical, so constraining them to vibrate and rotate as they commute linearly down a channel with dimensions small enough sounds more like the vacuum, one molecule at a time. We can't send Laplacian molecules down small channels even if we use just magnetic potential and charge to construct sheets.

 

[deferro]

So a tensor is a thing that let's you tally a cost, over two indices, say p,q, or s,t, or u.v.

The Coxeter graph describes a map into a mathematical space where primes evolve in a computational way; there is a recurrence and an encoding as permutations are made by rotations of groups of slices. Gravity does much the same thing to mass, where volumes have to be conserved, so that "space slices mass, or mass is the section of a slice of space, held by a pinning function, momentum or inertia". Force is like a fork that keeps mass doing things while space slices it.

And so with any alternating way to derive a product, such as the f and q turns you make with a pocket cube (a puzzle for rotators) there is a wedge that is a section of the space. If T the total number of permutations generated by (n)f, or (n)q turns, then: T1 -> T<sub>10<sup>7, after the Hamiltonian cycle that prepares the polytope in a prelaunch condition.

|H| is permute(fn,qn), so that p(f), q(fp) exist.
Then q(n) is primes in subgroup Aq; p(n) is primes over Af,q -> Tf,q; the indices can be raised and lowered for the tally, T.
At launch, there is the following condition in the T-registers {f},{q}:
{f1 + (3,2)t + (0)v},(q1 + (2,3)t + (0)v} = {(64)},{(34)}|T.

v is the volume of primes generated, or the V accumulator, t is the primes generated by the U(1) bundle on the trefoil, these are (3,2),(2,3) encoded as a subgroup; volume and density have generators which accumulate prime factors in rotations, and crossings, so the knot expands the toric bundle's volume/density of crossings.

... the pocket cube encodes a "primaverse", because, until you rotorate to 3, or 3x2 full.quarter, or 2x3 quarter.full, i.e. fq or qf turns so fq = 1, you are in the "universe". One more and you leave this breakpoint you get to u + 1 universe.</sup></sub>

 

[deferro]

Next, with the sphere, we want to generate primes, or map the toric bundle map B (from the wikipedia page on the Pocket Cube) to them with the same Hamiltonian U(1)xpermute(f,q(fp)), and the same spectral form for V,w.
Here w is a width for the Coxeter graph over a geodesic distance, or angular measure.

I want a 3-vertex geometry, so that a triangle on the torus {S(3,2)xS(2,3)|permute over t}; and a triangular sectioning with a volume form; sounds like the area of a spherical cap, with three equal sections removed (= r(3xd)) to form three straight equal length sides (ex3), so a bit of geometry for w.

Sectioning the cap should produce a regular shape so it has a equisectional area over the sphere S3; this is cube sectioning and dodeca- icosahedral volume forms against a sphere, generalizable to higher dimensions of toric bundles in the cap sections, angles and solid angles on the origin of hyperdimensional spheres with dimensions that are all factorisable with a 2-polytope coloring solution (in s,t on planet earth). Which might not have occurred to everone, but you gets what you looks at, regardless.

Having looked at a recent paper on Causal set theories and the structure of spacetime (my MIT assignment), I'd like to dissert on the role of CFT and AdS correspondence and the need to find a "mass" ansatz which leads to the highly-ordered Newtonian structure we observe locally.

Since the Heisenberg Lie group: H(1,C) is related in a "broken way", or isn't easily flattened or expanded (into a regular form) then p 'on a line' qn uses a robust or faithful map (see above) < H(1)> s.t. time-relations are canonical, or quantised. Classical complexed space (with time s.t. space OR time) is the S(t) of diff forms and operator algebras. BE stats, are a diff calculus (see OP) which need flattening against QT or quantum time.

The expectations seem to be that p,q of a singular theory is a limit of a canonical triplet, where [hat]p,[hat]q,[hat]r are mappings to a regular (geodetic) theory which is Lorentzian. SO(3) space in 1 time dimension is the unity of the Lorentz group s.t. H(1) descends (infinite recursively) into {SO(3) flexed by H(1)} and p,q unify.
The measures are quadruplets of seconds of spacetime distance (transformed to radians) meters of time (transformed to unitary quanta of distance) and Plancks of frequency in the time domain of matter (deBroglie spectral). Like the optical illusions we see in static graphs with contrast.