41

Physics, Astronomy & Cosmology / Re: What is the exact cause of the time dilation of the twin?

« on: 13/10/2024 04:29:56 »

Einstein's special theory is really like the law of thermodynamics: if your theory denies the reality of SR, there is no hope for it.

The twin paradox is a problem--a physics problem. But apply a bit of logic and it's obvious that the acceleration/continuous coordinate change is over a distance which is much shorter than the distances covered at constant velocity.

Hence the distance for acceleration(s) is negligible, hence ignoring that part of the journey has a negligible effect. Logic.

42

General Science / Re: Does All Of Mathematics Come Down To Simple Arithmetic?

« on: 13/10/2024 04:10:16 »

Arithmetic is most likely the first thing young children learn about numbers. One problem with this word is that it's a noun and an adjective.

Addition, multiplication, subtraction, division are all arithmetic operations on numbers.
Algebra is arithmetic, but not all algebras define subtraction or division necessarily. Some algebras are non-commutative, so not like ordinary "arithmetic" multiplication.

 All number systems are algebraic. Mathematicians generally discuss algebraic objects such as groups, rings, modules, categories . . .
I think even sets can be algebraic objects: set difference and intersection for instance. Or the interior, exterior and boundary of a set. It depends on the particular context I guess, but arithmetic is usually the simple end of it.

ed: after a bit of thought, the idea that a scientific theory can be reduced to a set of equations is perhaps an idea of simplification, but I don't know that that can be true:  without some contextual language a page of equations can't convey that much. Suppose Einstein had *simply* written his famous formula (a system of equations!) for gravity in one line and published it.

43

That CAN'T be true! / Re: Earthquakes and Presidents

« on: 07/08/2024 19:40:06 »

What's different about earthquakes and elections? One fairly obvious difference is that elections, at least in the USA, are held on a regular 4-year cycle.

Earthquakes don't do that, so I think it's safe to say the theory of presidential predictions is a simplified version of the earthquake prediction version. With a fixed cycle it's much easier to "distill" the factors with the greatest influence on the outcome and test the theory against historical results.

That last piece, the historical results, are less statistically meaningful also with earthquakes, because of the lack of longterm, consistently gathered data.

44

Just Chat! / Re: Political joke

« on: 07/08/2024 19:29:45 »

Trump Jumps the Electric Shark

. . . this is actually a plug for a new kid's game, in which a lifelike Trump doll jumps off a platform and hopefully avoids a tank with a realistic shark swimming in it!

The Trump doll is equipped with a speaker capable of 7.5 Watts peak output, with a volume knob and a separate dial for the level of crazy, from onset to fullblown dementia! Enjoy listening to your favorite quotes from the man Himself before sending him to his potential doom! Will he be eaten by the shark, or just get a realistic looking jolt?

Batteries not included.

45

That CAN'T be true! / Earthquakes and Presidents

« on: 06/08/2024 21:57:17 »

One Vladimir Keilis-Borok, a Russian geologist, developed a theory for predicting earthquakes which, when he collaborated with a colleague, Alan Lichtman, became a theory that predicts the outcome of US presidential elections.

So that has to be an example of looking for a pattern somewhere and then seeing the pattern somewhere else entirely. And it just so happens it's easier to predict a presidential race in the US than it is, still, to predict an earthquake, even in California.

Why though, to all the above? Why does it work or succeed, but fail in its original domain, this pattern-recognition theory, also why is Alan Lichtman a household name, but the other guy isn't?

46

Question of the Week / Re: QotW - 24.08.02 - What is the universe expanding into?

« on: 01/08/2024 00:32:42 »

The nice thing about logic is, you can look at it forwards and backwards.

In the latter case, the universe is contracting, and, Einstein's theories predict it contracts to a point. Except maybe not.

But the take away is that you imagine the quite large universe getting smaller. It doesn't have anywhere to leave an empty space behind it, that it might expand back into. Except that one is only true if there is a maximum of three spatial dimensions. Distances are not numbers, so they aren't much help here.

47

Just Chat! / Political joke

« on: 31/07/2024 21:59:19 »

Two of Darth Vader's minions walk into a bar.

The bartender asks, "So how's the planet conquering business, guys?"
One of the minions says, "Well, you know, gain a world, lose a soul" . . .

48

New Theories / Re: Could quantum mechanics be wrong?

« on: 17/07/2024 08:52:56 »

i cant help but spot the obvious here:

If you claim something is wrong with a theory, you're sort of obliged to point out what it is.

And pages later, you haven't. still. Nor has the team at the LHC, incidentally.

49

General Science / Re: How does ChatGPT work?

« on: 15/07/2024 00:32:05 »

What I think we might eventually see is an AI able to 'create' programs that it can prove are correct.

Why that is is because humans can do it already and all you need is logic. Induction and whatnot.
So to me, that will be down to being able to train some AI to use predicate logic and induction on some invariants, of loops and branches in the algorithms, however these are presented.

The faculty had cobbled together a "program logic" and embedded it as Java in C⁺⁺, as "comments" (erm, yeah another course I did). But an AI would in a sense, be that embedded code.

50

General Science / Re: what is Nothing or ?energy landscape.? or waves

« on: 13/07/2024 00:47:21 »

I also asked myself about the problem; we were not told why to use a  complex number for each transmitted or reflected wave, just that it works. And as the assignment, later covered in a tutorial, shows, it works. You get the right answer.

The solution doesn't need actual values for each amplitude, you just trot out A, B, C, . . . as needed. The problem is about the number of waves that exist, when light passes through a pane of "ordinary" glass. Other stuff in this course suggested that the solid state (any solid state) is not ordinary, not by a long shot. But that could be another story.

p.s. I think my use of "complex probabilities" is ok, 'cause I also call them probability amplitudes earlier on. If you're a student doing a solid state course, you might get a pass on that.

ed: I feel I should expound a little. What Eternal Student posted about wave numbers is the reason you can use a single complex number for each wave (or each reflection/transmission "event", or even "wave packet"). It's because you can expand a fourier series, and factor it to 'first order'. Use a unitary matrix exponentiation, if you must. But make sure it's Hermitian. We weren't told why to use single complex variables, because we were meant to know already.

51

General Science / Re: what is Nothing or ?energy landscape.? or waves

« on: 10/07/2024 22:45:56 »

There is a frequent and confusing misuse of "wave" when discussing "wave functions".

When I took a course at my local uni called Solid State Physics, we had an assignment that required the computation of the number of reflected waves for a "beam" of light incident on a pane of ordinary glass. The answer is easy if you just assign a complex number to each "wave"; this has the obvious advantage of being first order, and yet is a very good approximation (!). But the complex numbers are the probability amplitudes, for each reflection . . .

Is a light wave a complex number? No, the analysis of internal reflection at a barrier, for a pane of glass which has two external and two internal surfaces, is about complex numbers which say exactly nothing about what a lightwave "is". Maybe it's just an idea. And why complex probabilities? Erm, . . . linearity (?)

52

Physics, Astronomy & Cosmology / Re: why do a lot of people confuse between interference and diffraction?

« on: 23/06/2024 23:19:17 »

As usual, when I get access to a new AI model, I  check its understanding of topics I'm interested in.

AI with large-language models does not "understand" a subject. It is good at suggesting what words might be relevant and is trained to assemble meaningful sentences. That's about all it can do, search a large database which is a kind of dictionary, and string words together. It will make things up to try to impress you. (Ask that lawyer who lost their licence).

See if you can get one to understand French.

53

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 23/06/2024 08:21:10 »

Time for a tentative solution to the question what is the Hopf gauge?

It's a thing in a theory that gives the theory a place to start counting something else. Roughly.

Why graphs, ribbon graphs, are a way to represent this theory is buried in graph theory and homology and homotopy classes. Knots, and stuff like surgery of the kind that changes a symmetry, generally a topological one. It's all totally abstract, but, the idea is to have the right ansatz so you can plug in real particles with real paths; real external fields, and so on.

You get knot polynomials which represent real Lagrangian/Hamiltonian flows. And you get the need to start using new terminology. I can recommend a look at the Princeton Companion to Mathematics (google that), it's refreshingly honest.

I mean it doesn't just use the big words, it tells you what they mean and gives you concrete examples--it should explain better than I can what a gauge theory is.

54

Question of the Week / Re: QotW - 24.06.14 - Why do we retain maths ability better than facts?

« on: 11/06/2024 23:30:07 »

I think I have had similar experiences to others, in that I found logic and mathematics to be easier than say, microbiology or chemistry. I put it down to having to learn too many details with those last two, like having to carry around an encyclopedia in your head.

Mathematics can be about entirely non-physical things, which I guess I can assume straight away means more generality than the sciences restricted to such as biology or chemistry. I think that being able to write code has a lot more freedom, you can create whatever you like. It's sort of cathartic, I guess.

55

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 06/06/2024 03:04:40 »

Some comments on the way terminology has to evolve, and new mathematical kinds of objects are always subject to some overloading of meaning.

For instance, torsion is well understood in physics as the application of torque, in mathematics (group theory) it has a different meaning. Also, it seems to be down to whether or not polynomials exist in some mathematical theory which claims to represent reality i.e. the reality of entangled particles, the reality of measurements, etc.
The reality of a simple relation between sets and statistics that gives "reality" Bell's inequalities.
And a Nobel for proving, mathematically, that the universe is non-local. (??)

So the leading edge of all this Hopf algebra stuff, is a quantizable theory, this apparently means braid groups and abstract cobordisms between embedded links in R x R³. It has to preserve certain invariants, whatever theory emerges, and there is uncertainty about what these are, exactly.

56

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 07/05/2024 21:10:11 »

Ok, I have to include some actual stuff about what the big fuss is, why Penrose says much the same thing as, well a few people I suppose. The Hopf fibration does seem to be fundamental. Perhaps because the universe is just weird (except strips of paper are ordinary).

So from the Journal of Physics:

Hopf and the qubit?two-level quantum systems

The occurrence of the Hopf fibration in two-level quantum systems?systems that can be described using a two-dimensional complex Hilbert space, also nowadays known as qubits?was mentioned, e.g., by Penrose [36], and is described more extensively in [48]; so we can be brief and somewhat schematic about it here.

We take the two-dimensional Hilbert space just to be
with its standard Hermitian inner product ≔

. The states of the system are given by density matrices, i.e.,
Hopf and mechanics?the harmonic oscillator

It may surprise some that the simple two-dimensional isotropic classical harmonic oscillator provides an example of the occurrence of the Hopf fibration in physics. But indeed, its phase flow, restricted to a non-trivial energy shell, Hopf-fibers the latter! To see this easily, we scale everything such that the mass and the angular frequency of the oscillator become 1; the Hamiltonian then reads
whence the equations of motion

Employing the usual
Hopf and General Relativity?Taub-NUT space

Around 1950, G?del and Taub independently started the investigation of spatially homogeneous cosmological models in General Relativity. The ones constructed by them are space-times
having line elements of the form where the ωk are essentially left-invariant 1-forms on a three-dimensional Lie group and the product of forms in ds2 is symmetric tensor multiplication as usual. More precisely, , where

is some interval for τ, to be determined during the process of
Hopf and twistors?Robinson congruences

The concept of twistor can, according to its inventor, Penrose, be introduced in a number of ways, and all conformally invariant field laws in flat space-time can be reformulated in terms of twistors, generating new ways of looking at such laws; this has already been very fruitful in the past in many cases.

Clearly, this is not the place to give any of the different ways of approaching or using twistors. We rather pick out one way of representing a twistor space-time-geometrically and highlight
Hopf and Wignerism?helicity representations

In all examples discussed so far, the bundle, or total, space
was of immediate importance and thus in the forefront of the formalism; the base space

, on the other hand, did figure in the qubit case and is used in technical treatments of NUT space that work with the space of Killing trajectories. In the two examples to follow, it will be the other way round: it is the base space that is in the foreground, while the bundle space appears as derived, e.g., via a patching construction. The
Hopf and magnetic monopoles

This example of our collection is probably the first one that caught the attention of a larger number of physicists. As announced, the Hopf fibration comes here from a bottom-up construction.

In his 1931 article ?Quantised Singularities in the Electromagnetic Field? (magnetic monopoles), Dirac [19] starts out looking for a generalization of wave mechanics in which the wave function does not have a definite value for its phase difference between two points. In the 1970s, it was realized [50] that
Hopf and the Dirac equation9

The two-level quantum systems of Section 2 may be considered, in particular, as describing the spin states of a non-relativistic spin-

particle?i.e., one ignores its spatial degrees of freedom. If the latter is not done, one must write spinorial wave functions, subject to appropriate wave equations. In the relativistic regime, the relevant wave equation is the Dirac equation. We shall point out here that the Hopf fibration has significance also in this context. Most directly this comes about
Acknowledgements

I am indebted at least to the following persons for directly or indirectly educating me on matters included in this article: R. Beig, I. Bengtsson, P. Nurowski, R. Penrose, H. Rumpf, W. Simon, and A. Trautman. In particular, important information came from participants of a meeting in honor of E.T. Newman held in June 2001 at the Banach Center, Warsaw; I thank the organizers of that meeting for inviting me there to present a preliminary version of this article, for hospitality, and
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57

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 03/05/2024 05:44:38 »

It seems that the initial excitement about Erno Rubik's new puzzle, has seen the development, evolution if you will, of problems in the optimization or scheduling classes, or machine learning, in which, like chess or card games, the Rubik's cube and similar 3-dimensional puzzles are a good fit.

There are three main disciplines in which these kinds of puzzles appear, pure mathematics, or group algebras, vector spaces and modules, even as a good tutorial example of what they are. The second two disciplines are related, one is machine learning and optimization of solving algorithms in machina without human knowledge, the other is developing algorithms ad humana for humans to solve these puzzles efficiently, perhaps to study how well different people learn such things. Maybe the military can use it (just sort of kidding, there)

After all, demonstrating you can solve "the cube" is still a kind of test of your intelligence, and doing it quickly says you know some good algorithms.

58

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 02/05/2024 13:06:04 »

There are various techniques or methods that let you count permutations in a set, such as cycle decomposition and Stirling numbers, or polynomials of various types.

However, I can't help but notice that with a reduced set of generators and strings of compositions of two letters, counting is also much simpler.

Since a string like UR, is also a pair of group elements, there are inverses. So you also have U'R', U'R, UR', or a multiplicity of four. You only need UR x 4 because this is known about combinations, and by trivial induction that RU gives you four more. Then you extend this to 3-letter and 4-letter strings to derive some closed form expression or generating function.

59

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 30/04/2024 22:42:11 »

I've got  a nice result now for the chain-antichain decomposition. The key is to use a restricted--reduced--set of generators for the poset. Then each chain is just a string of alternating letters, these are accepted by a finite-state machine which constructs the poset graph according to string length and how many single repeats exist.

The graph is finite, so all strings which repeat are repeatedly--iteratively--mapped to this set of chains/antichains according to the structure of each string of characters.

For instance the string URURUR... will map repeatedly to the chain from (0,0) t0 (11,11). Why this is the limit depends on the symmetry of the group, or if you will, how the symmetries, extended into strings of length k (a group algebra), grade the set of permutations. Repeats like URRUR... will "lift" the string vertically, or alternatively I can push this into a third dimension in the graph, because It's my graph. I own this mother.

Because I have Z-modules, I have flat tori and these can embed in a topological space where the tori are linked together, but none of the connected paths in the graph are linked because they are all on the surface.
However there is a way to nest everything, and I don't know that it means a whole lot except in a topological, graph-theoretical context. It doesn't make the algebra any more useful or understandable, it's better as a nice linear map in at least two integer dimensions.

So the poset graph is embedded in a bundle of flat tori, which are linked together in an abstract space, i.e. are Hopf-connected, but G the cube group is not. Apparently.

ed apologies for that ownership claim. I own perhaps some of the ideas, not original really, but I don't own this group or a set of numbers. Those have been around for a bit longer than me, possibly even than the universe of matter and reality itself. Philosophical considerations aside, of course.

60

Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?

« on: 23/04/2024 23:19:22 »

I think I can say a few things about the modular structure of this graph.

In that, a free additive module is just the idea of repeated addition, a sequence like "+ 1 + 1 + 1 . . .". This, in modular addition, say mod 2, will mean you have the sequence 0 + 1 -> 1 + 1 -> 0 + 1 -> . . .
This means you can define a loop, you can identify the two points, {0,1} and make the distance between them + 1, as the loop so this represents the action of addition on both points, as a single point "{0,1}". So it maps a loop, representing addition modulo 2, to a set of points. This works for {0,1,2, . . .,n}.

So then you have 0 + 1 + 1 + . . . n times, = 0 (mod n). Yay. That's because 0 is the kernel of a function that maps to a homomorphism as the image of the next , or is it the previous? I need to figure that out, function (the homomorphism between additive groups) in the chain.

A Turing machine that models this is one that reads a 0 and writes a 1, or reads a 1 and writes a 0 ( adds 1 mod 2). So there are two read/write loops on a single state, and no halt condition.