Probability and Statistics

[Erasmus00]

A Rubik's cube has 6 faces that can rotate to any angle; there are an infinite number of partial rotations (in the slice group) for any single one of 6 faces.

 

While we can physically rotate the whole real physical cube, this isn't really what we mean by an abstract cube algebra. The structure space of the cube are states we get to by doing cube moves (90 degree rotation of the planes).

 

U(1) is a subgroup of the puzzle as a structure-preserving (diffeomorphism) algebra.

 

What diffeomorphism? What manifold is being mapped to what other manifold?

[Boof-head]

The structure space of the cube are states we get to by doing cube moves (90 degree rotation of the planes).

The structure space includes all the angles each face rotates through, during the rotations.

There are quarter and full twists, involved in the states. Each as you say, is a cube move but each requires that either 1/4 or 1/2 of the circle group of rotations is part of the structure.

 

The mapping is from the physical cube itself, to the color-logic or C-space the coloring makes. If you don't color the cube it doesn't work.

[Boof-head]

Back to the OP's original intent, after a bit of a path through what I would call a logic space.

 

But anyway, fair and balanced and what QM, the H word and probability mean to our epistemology.

 

Probable outcomes are about dependent and independent variables; these are often dice, coins, roullette wheels, or random number generators (more about these later). Where we use the concept of unitarity, or the number 1 we pronounce or give a verbal coloring to, as "wun", in most English-speaking countries.

 

This is congruent with the language's influence from Norman French, but it has a longer history - it was the first geometric symbol - the earliest tallying and counting alphabet consisted of marks in clay, made by a sharpened stick. This was shaped into a wedge

 

So the earliest language, with a recorded symbology that was used for systematic accounting and resource-sharing - taxing to provide for the future - was a wedge-product, an alphabet of discrete marks that eventually transformed into a meta-language - a state transition of a state transition grammar.

 

The grammar lost some of its original code; this was a numbering alphabet in base 60; the one we still use for time and angles of direction around a compass.

 

Probability is the domain of large numbers - the probability you can color a cube with colors up to 6, is 1. If you divide the faces of each cube into partitions it changes, but is still congruent with a finite result - if you can rotate these color-partitions into equiprobable maps that correspond to a fair-color map, you have a solution potentially because of slice groups.

 

Quantum states involve two variables - a wavefunction has 2 equiprobable states for any quantum 'particle' in the same sense as a colored partition, or a map of them on a colored cube.; here Rubik's cube is a mechanical solution for a fair color-map.

 

The mechanical solution for a wavefunction requires a formulation that corresponds to "one event" - here the time the event occurs is the probability, OR, the position it occurs is. These views as in classical ones, are interchangeable, but in fundamentally different physical ways. Space is quantized in QM, and time is linear - we assume the latter, since there is no evidence it isn't to say otherwise - time is Minkowskian in the local EM frame, it's as flat as the side of a coin.

 

Say you have some coins, which are different, say copper and silver, or C and S.

Now you put them in a bag you can reach into, without seeing, so you can randomly select one - so far so probabilistic. If you put 2 each of the S and C coins in to start, their states - when you select the coins or measure the probability-space - are entangled with the measurement - in space and time, but outside the bag.

 

If you select a C, you have 1 left in the bag so there is now 2/3 probability that you will select a S next; if you multiply the number of coins - scale the space - this changes linearly as you do, until it returns to 2/4 again. In fact this 'return', given a finite set of 2(S + C) once you have 1 coin is only seen at infinite set size.

 

Now if you distribute the coins, say by giving them to a large group of people, then you sample these subsets by asking each person how many coins they have, then selecting just the 1 coin again, what is the probability that you know the "next" coin?

 

What if the person lies, how can you find out? Can you prepare something beforehand that gives you an "extra"?

 

Yes, if the coins have QM properties - if they are quantum states.

[Boof-head]

So restarting from the state space of coins in containers which are a big bag of them or a large collection - of two states which is also a dependence, a fundamental one - start with the usual simplification, a single coin.

 

Not assuming anything about a coin except the two sides are different, there is exactly 1/2 for each state in the space. With 2 coins there is 1/4 for each state. This expansion is why 2 is important, a bipartite system is 'bigger' because it has more entropy.

 

With 2 coins that each have 2 independent states, you color them to separate them; separability is guaranteed with coins. So with S + C coins there are ShCh StCh ShCt StCt possible selections; tossing them is the mechanical 'probability generator'.

 

In principle you can build a machine that will guarantee a state by tossing coins - or with 2N coins will generate a pseudorandom sequence, the mechanical gotcha with construction (which includes constructing a quantum coin).

 

if you select, as before one of the above outcomes, what you do is find a predecessive state, you remove a state from the denominator of the probability measure, like this:

[math] (S | C) / (S_{t,h} \oplus C_{t,h} - u) [/math]; u is a measure operator that reduces the number of available remaining selections by 1 of them; the numerator is the number of coins, or "C or S" a Boolean, the subscripts are the state index for "copper or silver".

 

A quantum-mechanical coin can be copper and silver, and also heads and tails, but not when we select one. The quantum coins in a bipartite state (which can be a single one) can be entangled, the phase space has a mechanics which is time-independent the same way coins in a bag are, until you select one; the selection and preparation for it are independent, except in QM the last part is not trivial.

 

Ignoring this problem means you "just" need the measurement operator, which is a square probability amplitude. Since a square (or a square matrix), has a diagonal the square root of two, there is a non-enumerable quotient in this space; this is the key to getting something for nothing, because a unitary state has an exponential 'hidden' phase [math] e^{i\phi} [/math]

This is distributed over the two in a bipartite state - the key is preserving a phase despite measurement, or "not collapsing" the state.

 

In fact there is never any quantum collapse, this is a consequence of preparation and measurement.

[Boof-head]

Returning to the "ersatz" for a color-tensor, which is a 2x2 sub-matrix on a Rubik's NxN cube, for N = {2,3,..,7} [here, I mean for the projection P1, a single face]

 

The way to think about what a tensor is in classical space, is usually as a representation of two states, labeled (0,1), a numeric basis for them like sides of a coin, or like binary; sticking with the index notation that the color at the centre of an odd N cube's faces gives any face the index (00), we have that c, the color is background for the 2x2 facelets, say in the upper right corner - we apply a right-hand, positive sense to this subspace like the notation for quadrants of a circle.

 

This means we can label the facelets of corner pieces with the same 'metric', which for a quadrant is from the following set: {+ +, + -, - -, - +}, an orthogonality measure of this subspace.

 

For the connected region for a 2x2 matrix (a submatrix of the 3x3) the symmetry is generally written as [math] \Psi [/math], which equates to [math] \big\langle |A| \big\rangle = \big\langle A \big\rangle = \alpha |0 \rangle\, +\, \beta |1 \rangle\[/math]; [ed: correction made] where the symbology means the same: "|" means an area, "<,>" are edges that correspond to the inner and outer edges of a sliced cube. "Inner" can mean "outer" depending on the basis being used. Mixing these is a function of preparing states.

 

Preparing a state implies arbitrary rotations, or exchanges over the space; swapping corners or edges has a general "swap" operator, that is given 2 of each, and exchanges their position - this will invert something. If you swap the two edges in a 2x2 matrix on the cube you invert the 'hidden' colors, which are 'over the edge'. Thus "inner edge" means hidden color. We are in a projection that includes a single face and 9 partitions in a 3x3 matrix.

 

The "wavefunction" or [math] \Psi [/math], has 2 components with abstract color coefficients [math] \alpha, \beta [/math]. These are 'fixed' because the colors are, but if the cube had a countably infinite number of colors it would look like a quantum state - even for 2, 3 etc or 'small' N, since we can use a 'quantum color'.

The requirement would be that swapping still inverts color 'phases' as with fixed coloring.

The evolved state looks like a color in the quantum version when you 'look' at it; it will be a function of "color" that has a probability of being 'seen' given the color-map, which now resembles a cylinder between two 'sphere states' as in P-space.

 

The classical space has color vectors that still behave like vectors, these are color-pairs which are matched and adjacent (conjoint); pairs which are disjoint are only possible between the null-color c at the center or pivot and its corresponding corner facelet.

 

The color-dimension for each of the facelets in the 2x2 subspace are: for the center, 00, for the corner, 11, and 01,10 for the edges. The null-color is 0, the edges have 1 extra, the corner has 2 so the indices represent "how many" hidden colors; swapping edges will 'flip' this index from 10,01 to 01,10 or back again - if one face of each is the null-c or color of the projected space.

 

The vectors have to be Hermitian for QM, such that a swap + unswap is Abelian, or:

[math]\;\; VV^{\dagger} \,= \, V^{\dagger}V[/math] which says multiplying V by its transpose is the same as multiplying the transpose by V. Something that needs to be demonstrated with the cube (but is true in QM and Hilbert spaces).

 

On a fixed-color cube, the edges 'take' [math] \alpha, \beta [/math] as their hidden colors disjointly, the corners have both (conjoint union). We have the Boolean 'swap' or inversion, conjunction, and disjunction; all of logical "colors", as vector pairs relative to the null-c one at each central pivot.

[Boof-head]

It's easy with tensors (at least when you can reduce a set of "vectors" to a binary rep).

 

The states '0' and '1' in binary logic always correspond to a real rep (as charge in capacitors/transistors) in some physical "circuit" or other. The generality or 'universality' of Boolean logic is the reason we have built electronic computers, that realise these states by 'coloring' capacitors with real charge, and 'decoloring' them; both cases are the physical basis for the logic. Logic is independent of physicality, in the sense we now have faster circuitry than we used to - so the logic now 'clocks' at a faster rate - rate has a thermodynamic limit, constrained by real time and the thermodynamic limit of any material used to construct a machine.

 

Chernobyl, for example, is a signature example of "thermodynamic limit" being exceeded - the energy (of the process) could not be limited by the physical machine - result catastrophic failure; the algorithm halted. Machines halting is a very prosaic part of our modern lives (esp. for Micro-snot users).

 

The vector space has these 'digits' in it in a binary machine; these are generally taken in pairs and switched together, but also singly. The only 'useful' thing that can be done with "one bit" is inverting its state: 0 -> 1; 1 -> 0.

 

Otherwise you switch 2 bits, to obtain either a sum (OR or XOR), or product (AND); Negation is a locally trivial operation, the 2-dimensional transforms that obtain sums and products, represent a 'computational flow', which, since a path exists physically, will "yield" a result after some time t. The t-factor depends on thermodynamic scales, the computation depends on how many bits get inverted, summed and multiplied - this is a path through an abstract space, which is independent of the physical real machine that 'constructs' a real representation.

 

On the colored, sliced cube, the vectors are 'positions' over 2x2 sub-matrices, into which 'colors' are rotated. When these are the same as the central null-color at (00), you have pairs of colors, the space is orientation or position-dependent. To fit a tensor algebra into this, you have to develop a consistent formulation for each vector.

 

Since the edges are distinct (2-colored) these represent the states (10) and (01), which says: "the top edge-color is to the left, the right edge-color is to the right". It's ok to mix top/left, and right/right, since these vectors have 'mirrors' around the projected face; then we extend the two (positive, upper-right) edges to the other two, by multiplying them by -1; we get (-1)(01) -> (0-1), and (-1)(10) -> (-10); there are 4 hidden colors now.

 

These are the basis states (10), (01), (0-1), (-10) for edges, when they are swapped around. We can change this basis at will: for instance the rep (1), (2), (3), (4), is no problem as long as we can map it back to a standard basis; in QM this is [math] |0 \rangle_a |1 \rangle_b \, \,=\,|01 \rangle_{ab}[/math]; for bits a,b. These are usually also labeled "Alice and Bob".

 

They correspond to (10)a (01)b = (11)ab in classical bit terms; the sum, in XOR of two states, requires 2 XOR ops each 2-dimensional, or a 2-level circuit; in that sense you are XORing a,b to get a 2-dimensional result, which 'erases' the 0 states. This is equivalent to erasing (00), so that only (01, 10, 11) are left and implies (00) is a 'ground state'.

[Boof-head]

So, far all of the algorithmic languages I've invented for this 'logic-space' except for the QM notation, are naive, in the sense they will need to be revisited so they correspond to accepted matrix arithmetic, in which rows and columns exist together.

 

The 2x2 matrix is square, and rows and columns are equivalent, so that 2 rows, or 2 columns = 2x2 sq matrix. The t + t = t x t formula of a time-ratio, which has exactly one solution, t = 2.

This correspondence is universal, also it represents a universal principle which applies to cosmology, GR/SR, QG theories, and spinfoams in spacetime - the vacuum is a spin liquid with a background structure, in which this universal formula must exist, or there wouldn't be one. There would be an irrational result, non-enumerable on any UTM, or any logical machine, including a universal one we might design or construct in principle.

 

An abstract version of the universal Turing machine is the universal function, a computable function which can be used to calculate any other computable function. The utm theorem proves the existence of such a function.

 

When Alan Turing came up with the idea of a universal machine he had in mind the simplest computing model powerful enough to calculate all possible functions which can be calculated. Claude Shannon first explicitly posed the question of finding the smallest possible universal Turing machine when in 1956 he showed that two symbols were sufficient, so long as enough states were used. Shannon himself proved that it was always possible to exchange states by symbols.

 

After some time, the smallest known universal Turing machine was due to Marvin Minsky who in 1962 discovered a 7-state 4-symbol universal Turing machine using 2-tag systems. Applying Shannon's result to Minsky's UTM upon conversion to a 2-symbol machine Minsky machine would require 43 states.

 

Other smaller universal Turing machines have since been found. If we denote by (m,n) the class of UTMs with m states and n symbols the following tuples were found by Yurii Rogozhin in 1996: (24, 2), (10, 3), (7, 4), (5, 5), (4, 6), (3, 10), and (2, 18). In 1985, Stephen Wolfram conjectured a 2-state 5-symbol universal Turing machine. This conjecture was proved by Matthew Cook working as a research assistant to Stephen Wolfram. The model, also known as Rule 110 Elementary Cellular Automaton had, at the time, the smallest product (2,5)=10 of any known universal Turing machine. According to Wolfram other smaller UTMs should exist and he proposed a 2-state 3-symbol Turing Machine as a candidate. On 24 Oct 2007, Wolfram announced the Turing equivalence of the system had been proven by Alex Smith -- an undergraduate studying electronic and computer engineering at the University of Birmingham -- responding to a contest established by Wolfram.[1] However, on 29 October 2007 Vaughan Pratt of Stanford University announced that he discovered a flaw in the proof.[2] Wolfram Research disputes Pratt's interpretation.[3]

Universal Turing machine - Wikipedia, the free encyclopedia

[Boof-head]

 

Without further ado I'm going to attend a lecture by Seth Lloyd, which unfortunately I will need to imagine. But he has left some notes on the handout shelf for the grads at MIT; so why not see if it makes sense? Is he introducing these ideas in the sequence implied? I suppose so, so does this implication say anything and is it ok to go backwards over them, to "descend recursively" so as to converge on something?

 

Well, maybe; let's see.

 

The above handout is a simple diagram of a circuit in 3 states. The last has Hadamard input and output gates in 1 dimension, the inner circuit has 2 inputs which means they are the 2-dimensional transform of the outer Hadamard product.

 

So far so simple - the H gates are acting like quantum inverters, the inner gate is a switch, controlled by one or the other input. There are 2 outputs which represent an outer product; there is a phase-difference between the 'free' output and the gated one with (an) H, which is a function of the inner hidden phase - the ground state - [math] e^{i \phi} [/math].

 

A color-analog, using the usual model means designing a MkII colored cube. A computer model will work or you could design one that changes colors (over 6) per facelet electronically, so it corresponds to valid moves.

Then a scrambling exists that in principle will 'oscillate' each outer facelet into an indefinite mixed color except for the fixed one which stays in the ground state or [math] \phi[/math]-colored.

 

P.S. I have also uncovered a nice analysis of another version of this 'fixing', with the void-cube which has no central colors or even pieces. This constrains the space again in a way that uncovers something else about Rubik's original design. The void-cube is another inductive step.

 

But it's all in French so it needs translating; say into algebra.

[Boof-head]

The CNOT gate controls the inversion of one of the inputs using the other. The middle diagram is the first step toward the Hadamard CNOT at the bottom. The professor is reviewing at this stage of the course, the circuits introduced at the beginning, all couched in Boolean logic and FETs, where information disperses, or is erased at a cost of maintaining it (against dissipation). It costs something to copy information, in the same sense it costs something to build a circuit.

 

It's a space problem in EE and digital fabricated circuits, and a thermodynamic one; in QM this switches around, in a way that reveals a deep 'structure' in information/representation that Shannon blithely uncovered and we now use to calculate, as Seth Lloyd does, a limit or bound for the amount such spaces can contain; we call them black holes.

 

Another detail about sliced cubes and rotations, the Moebius loop is part of the logic.

This is because you can rotate two slices at a time (antisymmetry); flipping orthogonal edges means they travel around a strip which is 'non-orientable' (not enumerable, or indefinite), but does have an inner orientation - the edges which are locally adjacent.

 

So that x,y cube edges or (0,x),(y,0) pairs represent the edges of a strip (of paper), the surface is represented by the center and corner, so is 'central'; the twist in the loop is represented by the slice-exchanges.

 

This is easy to project topologically since you can rotate a slice so that two edges are "half way" to a quarter turn; the center has 3 edges left, since there are 2 slices left (in the nullspace).

Rotating another slice this way is only possible in the same plane, so that the center has only 2 edges adjacent. This maps to the construction of a Moebius loop from 2 antiparallel directed sides of a square, and so the trefoil.

 

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram

 

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle [math]S^1[/math] with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or [math] \mathbb{Z}_2 [/math]) bundle over [math]S^1[/math].

Fibered knot

 

 

A knot or link K in the 3-dimensional sphere [math]S^3[/math] is called fibered (sometimes spelled fibred) in case there is a 1-parameter family Ft of Seifert surfaces for K, where the parameter t runs through the points of the unit circle [math]S^1[/math], such that if s is not equal to t then the intersection of Fs and Ft is exactly K.

 

For example:

 

* The unknot, trefoil knot, and figure-eight knot are fibered knots.

 

* The Hopf link is a fibered link.

 

Fibered knots and links arise naturally, but not exclusively, in complex algebraic geometry. For instance, each singular point of a complex plane curve can be described topologically as the cone on a fibered knot or link called the link of the singularity. The trefoil knot is the link of the cusp singularity [math] z^2 + w^3[/math]; the Hopf link (oriented correctly) is the link of the node singularity [math] z^2 + w^2.[/math] In these cases, the family of Seifert surfaces is an aspect of the Milnor fibration of the singularity.

 

A knot is fibered if and only if it is the binding of some open book decomposition of [math]S^3[/math].

From Wikipedia, the free encyclopedia