@mchambin, I notice a significant loose end in your Maths Quiz!
I say this as an English Mathematician:
Surely I have won the Maths Quiz? You will know the Lion by his claw. 😆
Aaa... the good old CNN Black Hole... the one that swallowed flight MH370?
BUT... how small can a black hole be? What would the event horizon for such a "smallest" black hole be if its the same size as the unit of measurement for gravity? It means that there must be smaller sub units of the Planck Length. As in a microℓP, huh?
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It is fun to see Euclide approach. I was not aware of variants in the Euclide theoreme proof about prime numbers infinity. I wonder wether these are alternate Euclide works or alternate translations / interpretations of ancien Greek texts.
It seems rather incredulous to me that in GR the vacuum energy is the equivalent of 3-4 hydrogen atoms per m^3 and in the QM view, it’s 120 orders of magnitude greater and particles pop in and out of existence in a ‘quantum foam’.
To add to the mystery, QM theorists state that ST is quantised in units that are many, many orders of magnitude smaller than an atom.
I can’t help feeling Einstein was led by intuition, thought experiments and logic with mathematics used as the tool largely at the end of the process to prove his ideas while the QM world, (which we know works), has in some aspects been led by mathematics and the reality then described by the resulting outcome. Both theories largely work very well but it seems there remains an area where they profoundly disagree.
No wonder there is a crises in physics and cosmology!
To add to the mystery, QM theorists state that ST is quantised in units that are many, many orders of magnitude smaller than an atom.
I can’t help feeling Einstein was led by intuition, thought experiments and logic with mathematics used as the tool largely at the end of the process to prove his ideas while the QM world, (which we know works), has in some aspects been led by mathematics and the reality then described by the resulting outcome. Both theories largely work very well but it seems there remains an area where they profoundly disagree.
No wonder there is a crises in physics and cosmology!
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Could it be like Newton - Ok until Einstein cropped up. Perhaps he should have said physics doesn't throw dice.
The aether did well for a long time and diffraction models have worked well too. 😉 I have always wondered why the aether should have a direction. LOL due to debate via letter in wireless world some years ago.
Plank constant. From what I can gather derived empirically. Assume infinitely small and work from that appears to have been the technique used.
Relativity - quantum mechanics. In terms of things than can be checked which one has the most lead in it's pencil. "Undetectable" particles that pop in and out of existence but have an effect ??? Things can only be observed etc. It's possible to infer all sorts without having the ability to check - pretty popular in the astro world.
Could this huge difference be the first instance that something somewhere is wrong with quantum physics. Just that it works in some areas to some limits doesn't mean it's ok where ever used - eg Newton and similar in a number of areas.
Einstien mentioned the constant in his photoelectric paper but seemed to be extremely unhappy with aspects that followed. Perhaps he is correct.
I am utterly ignorant of these things and will admit it publicly. I know a very small bit about amplifiers and that’s it.
However, doesn’t’ultra small particles that we will never be able to detect but have a huge impact on the world around us’ and BTW have energy densities of billions of joules per m^3 which we also can’t exploit sound like religion?
I think physics has lost its way with this stuff. Ed Witten is undoubtedly one of the most brilliant mathematicians around, but he’s spent his whole life chasing this and come up with little but some magnificent equations.
However, doesn’t’ultra small particles that we will never be able to detect but have a huge impact on the world around us’ and BTW have energy densities of billions of joules per m^3 which we also can’t exploit sound like religion?
I think physics has lost its way with this stuff. Ed Witten is undoubtedly one of the most brilliant mathematicians around, but he’s spent his whole life chasing this and come up with little but some magnificent equations.
An easy phrase to write, but a difficult one to explain!
Recursive - characterised by recurrence or repetition.
Apparently, the uniformity of space-time is a consequence of its recursive generation. 😵
Read on for possible illumination: https://ui.adsabs.harvard.edu/abs/2015APS..APRC13003M/abstract
Let's look at primordial black holes which were formed when the matter density exceeded nuclear levels within the first microsecond of the big bang.
https://www.scientificamerican.com/article/quantum-black-holes-2007-04/
Recursive is a bit more than iterative. At least in programming.
Iterative calculation of factoriel n, is doing: 1 times 2 times 3, times 4 ...up to times n.
Recursive calculation is doing: n times factoriel ( n -1).
It looks the same but is not executed the same way.
Iterative calculation of factoriel n, is doing: 1 times 2 times 3, times 4 ...up to times n.
Recursive calculation is doing: n times factoriel ( n -1).
It looks the same but is not executed the same way.
LOL The wiki is interesting. One meaning said to be informal
Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be 'recursive
That in software terms can mean a procedure that calls itself.
In mathematics and computer science, a class of objects or methods exhibits recursive behaviour when it can be defined by two properties:
- A simple base case (or cases) — a terminating scenario that does not use recursion to produce an answer
- A recursive step — a set of rules that reduces all successive cases toward the base case.
Fib(0) = 0 as base case 1, Fib(1) = 1 as base case 2, For all integers n > 1, Fib👎 = Fib(n − 1) + Fib(n − 2).
the formal definition of the natural numbers by the Peano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."[2] By this base case and recursive rule, one can generate the set of all natural numbers.
It's example of a recursive step is a bit stupid but the reduces indicates what it means.
I just love the comment
But even if it is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of the procedure; this requires some administration as to how far various simultaneous instances of the procedures have progressed. For this reason, recursive definitions are very rare in everyday situations.
I think that sentence needs a re-write, AjohnL, however it's really recursive space-time I want to get my head around. 🤓
I suspect I have zero hope of understanding this 21 page pdf: https://link.springer.com/content/pdf/10.1007/s00006-022-01235-x.pdf
Steve should take a look, simpy 'cos it mentions the implications of conservation laws by Noether’s theorem!
Amalie Noether was a great Mathematician.
https://en.wikipedia.org/wiki/Emmy_Noether
I am still waiting for the Maths Quiz prize from @mchambin.
I think AjohnL is Total Troll. Sorry.
I have no time for such people. 😳
https://en.wikipedia.org/wiki/Emmy_Noether
I am still waiting for the Maths Quiz prize from @mchambin.
I think AjohnL is Total Troll. Sorry.
I have no time for such people. 😳
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Wot r u gettin' at, Bonsai?
A photon has energy, but no rest mass.
A photon does, however, have relativistic mass - mass when it is in motion, as opposed to at rest.
A box full of light would be more massive than an empty box by virtue of the energy of the photons.
You won, no question about it.
As a bonus, can you find how many n decimal digits numbers there are with descending digit value order I mean numbers like:
86643, 920, 9998663221.
It becomes pretty difficult for n over 6.
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Since, energy and mass are equivalent, a consequence of the famous equation E = mc^2, photons can be said to have a mass equal to hf/c^2 and a momentum hf/c. The term hf is the energy of the photon. This can be used to derive the equation for the Compton Scattering Effect. The derivation is based on the conservations of energy and momentum from which 3 equations can be obtained.
The above if one example of the so called duality of matter.
The above if one example of the so called duality of matter.
Just wondering about the very early universe when only photons existed AFAIK (see Stephen Weinberg 'The First Three Minutes'). If this indeed was the case, gravity could not have existed, since photons are massless although they may have relativistic mass.
Do I understand this correctly? Mass, and therefore gravity, must have arisen only once the early universe cooled enough for more exotic particles to have emerged that then gave rise to mass and gravity.
I read some time back that the zero-point energy density of the vacuum is extremely large and would require a cosmological constant (like the one introduced by Einstein) which is larger than to be expected by about 120 orders of magnitude.
This “cosmological constant problem” remains unresolved.
The best known example of the consequence of zero-point field energy is the attractive Casimir force between two perfect, plane, parallel mirrors.