joigus

Enough said.

Virtually everything. The most rigorous proofs are generally overlooked, like those based on epsilon and delta to prove existence of limits or continuity, etc. Multivariable calculus is used a lot. Also infinite series, limits, derivatives, integrals, improper integrals, complex analysis... The whole shebang!

That's what I said. Are you repeating what I said? Only the property is not unique. Any tensor product of 1-covariant 1-contravariant tensor also has that property. And same with epsilon tensors (in that case the components are 1, 0, and -1). The Kronecker delta \( \left. \delta^{\mu} \right._{\nu}\) is an isotropic tensor. The Kronecker deltas \( \delta^{\mu\nu}\), \(\delta_{\mu\nu} \) are not. On the other hand, from your document (ineq. 1), it does not follow, as you say, that, \[ \alpha\beta -\left| \boldsymbol{\alpha} \right| \left| \boldsymbol{\beta} \right| \geq 1 \] (expressed in a lighter notation). Your ineq. 1), e.g., would be, \[ \alpha\beta -\left| \boldsymbol{\alpha} \right| \left| \boldsymbol{\beta} \right| \geq \sqrt{ \alpha^2 - \left| \boldsymbol{\alpha} \right|^2 } \sqrt{\beta^2 - \left| \boldsymbol{\beta} \right|^2 } \] The above expression does not follow, as this counterexample shows: Pick \( \alpha = 1 = \left| \boldsymbol{\alpha} \right| \); \( \beta = 1 = \left| \boldsymbol{\beta} \right| \), but, \[ \alpha\beta -\left| \boldsymbol{\alpha} \right| \left| \boldsymbol{\beta} \right| =0 < 1\] So you're wrong here. Other mistakes have been pointed out to you repeatedly. Time to go back to a relativity book and do the exercises.

https://www.perimeterinstitute.ca/video-library/collection/2015/2016-complex-analysis-tibra-ali I generally recommend Perimeter Institute Lecture Series. I haven't followed this particular one, but quality is quite good at PI.

Indeed. Maybe we all are tools or someone who's someone else's tool. With no overriding handler of all tools. Paraphrasing @dimreepr, imagine that...

🎬 To waffling with waffling. peheh, pahah, poohooh. To me, to you, to us, is not significant. Unless disclaimers, caveats or qualifications are applied. 🎬

In fact, a Kronecker delta that is twice covariant or twice contravariant is not an isotropic tensor either. It must be 1-covariant 1-contravariant. Also arbitrary tensor products of 1-covariant 1-contravariant Kronecker deltas is an isotropic tensor. Arbitrary products are not. That's because contravariant indices transform with the inverse matrix with respect to covariant ones (that's why they're called "contra"). If you multiply twice by the same matrix you don't get back to Kronecker deltas. You must go carefully through all these checks in order not to make elementary mistakes. It's a natural rite of passage. The literature is full of mistakes of this nature. Not in the really prestigious books, of course. No. Mathematical physics is expected to cater to physics. Mathematics doesn't need any imput from physics. Mathematical physics is expected to be self-consistent, and further, it is expected to comply with what we measure in the laboratory.

invisible braces: \[ {\left. \delta^{\mu} \right. }_{\nu}\]

Funny that you think that, as nobody else does. Without going into details, it's quite clear for anybody who's worked with tensor calculus for some time that the most likely thing going on is that you're confusing invariant properties with coordinate-dependent properties, and mixing them all up in one big mess. Some tensor identities can be proved by appealing to some tensor being zero in one particular coordinate system. Then it must be zero in all coordinate systems. Conversely, non-zero in one system <=> Non-zero in all. On the contrary, the Christoffel symbols can always be chosen to be zero in one coordinate system, but non-zero in infinitely many other coordinate systems. Playing with these two properties facilitates some proofs, but you must know what you're doing. Handling index expressions without any care of what is a tensor and what only holds in one particular system is the wrong way to go. It is a real pain to go over every step of a calculation that somebody only too obviously did wrong, because you have proven the theorems forwards and backwards and gone through all the examples. A tensor being diagonal, e.g., is not an invariant property under SO(3) or O(3). On the other hand, tensors like the identity \( {\left. \delta^{\mu} \right. }_{\nu} \) or \( \epsilon_{\alpha\beta\cdots} \) are called isotropic tensors, because they look the same (have the same components) in all coordinates systems. So you cannot safely assume that a diagonal tensor takes part in any tensor equation. "Diagonal matrix" makes sense, meaning "a matrix that looks diagonal in a particular base". "Diagonal tensor" does not.

I would have to be on top of the hill to look down on others here to evaluate them. I'm not in such position. How smug would I be if I did? I stand by my words: A superb explanation --especially considering the limited amount of time and manoeuvre we all have here-- by a person well versed in mathematics who has all my respect. MigL's explanation was also very helpful, although in a very different style and spirit. As to your kind offering of starting another thread, I'm not so interested in judging people as in examining ideas, and trying to understand some of the most difficult ones. But you're free to open that thread if you want. Here's smiling at you

Easy, because as @swansont told you, you're going around in circles. \[ \frac{\sqrt{FE_R} \left( \sqrt[4]{FE_R} \right)^2}{\frac{E_R}{c^2}} = \left( FE_R \right)^{\frac{1}{2}+\frac{2}{4}} \frac{c^2}{E_R} =\] \[ = Fc^2 = \frac{G}{c^2}c^2 = G \] You derive guess an equation from your definition. You substitute your definition, so you get to an identity. Doesn't matter that your definition dimensionally has no relevance. And your \( g_\text{photon} \) has the funny dimensions of (length)3/2(time).1.

\[ \frac{\sqrt{FE_R} \left( \sqrt[4]{FE_R} \right)^2}{\frac{E_R}{c^2}} = \left( FE_R \right)^{\frac{1}{2}+\frac{2}{4}} \frac{1}{c^2} =\] \[ =\frac{FE_R}{\frac{E_R}{c^2}} =Fc^2 = \frac{G}{c^2}c^2 \]

If I may say something... I was aware that @wtf was giving a superb mathematician's exposition of the topic, while @MigL who had had some previous experience with the OP, was quite deliberately trying to dumb it down. It was fun seeing you interact. But your effort is not in vain, wtf. Thank you. I appreciate it.

\( F= \frac{G}{c^2} \) has units of (mass)-1x(length) Energy has units of (mass)x(length)2x(time)-2 So \( \sqrt[4]{F\times E_\text{photon}} \) has dimensions of (length)3/4x(time)-1/2. So that's a non-starter from dimensional analysis alone. Sorry, I honestly thought you were joking in the Physics section. I immediately removed the neg reps.

Happy birthday and many happy returns!

You are serious, then? I couldn't believe you were serious.

As long as it's just a tool. Now that I think of it my metaphor of the root and the seed was not particularly illuminating. LOL Edit: x-posted with iNow.

(my emphasis) This sounds ridiculous to me. Eukaryotes throw away a lot of the code (introns) by splicing, and some bits they keep for "creative" combinatorics, because the stop codon has a variable locus depending on the initial codon for translation, after transcription --if I remember correctly. The Duchenne muscular dystrophy gene being one of the closest to having record-breaking code redundancies. Admittedly, a deleterious gene, but eukaryotic splicing generally is very wasteful in terms of codons. Eukaryotes have virtually no limitation in that regard. A "sub-routine" of the transcription process could coincidentally be similar, but the whole architecture? That's basically why a tomato has about as much genetic code as any of us. Another example is the process of proofreading, in which for some polymerases goes back and forth. Why would you delete code only to rewrite it again? Nothing like that happens in engineering, AFAIK. So to me, the argument does not even stand to reason. But let the experts speak. I'm always willing to learn from people who know much more than me.

Silly me. I missed those.

If I were to try a model for what you seek, I would look for inspiration in ideas like: https://en.wikipedia.org/wiki/RNA_world. https://en.wikipedia.org/wiki/Ribozyme From the point of view of theoretical physics: Microscopic information is conserved. So information does not need to be generated. It's there. So-called coarse-grained or macroscopic information is not conserved, however, and the 2nd law of thermodynamics tells us it always decreases in a closed universe. But in open systems subject to external fluxes of energy it is known to give rise to order formation, or clustering of macro-information, if you will. So the 2nd law can "go backwards" locally, so to speak. If you add a principle of replication (structures appear and disappear, but give rise to structures similar to themselves) but with small differential changes between replications, you've laid the groundwork for explanation of this astonishing illusion of design without a designer. So we actually already know the answer to,

My French is a bit rusty, but I can tell you as much as this: Fire does not involve the disappearance of matter. The concepts of an infinitely big Mandelbrot set and and infinitely small Mandelbrot set do not make mathematical sense. The Mandelbrot set is invariant under discrete "zoomings". The sentence, "The only way to represent the infinity of a material thing is the circular shape, where the beginning and the end merge" is ambiguous enough so as not to make any mathematical sense.

A part of me wants to believe that religious types who drop by have a part in them who is desperate to be won over by a set of more solid arguments... or perhaps more positive, constructive doubt. It's a bit disappointing, rather than offending, when they turn to calling you names. I notice that frequently people with strong opinions rarely drop them in front of you. We all are hardwired not to lose face. That's probably because our competitive primate nature has grown ramifications into language itself. Ideas are more like the proverbial seed that our present interlocutor has mentioned, rather than roots that try to break through the ground. They normally sprout when you're not looking.

Oh, we know this is trash-can material from day one, don't we? Like most other users I just wanted to do my part in bringing it out. True colours showing. We've got the full spectrum, from kefir to non-clay bricks, plus tidbits of old-time religion. All topped with insult, instead of arguments. I think I'm done.

You see genius in a seed because there are billions of years of incremental improvement in this marvel of coordinated chemical actions. Mountains of evidence is precisely what's helped us understand what a seed is. When it develops, it recapitulates the history of the Earth, so in a way, a seed has chapters of the history of the Earth written in it. It took centuries of human curiosity to end up in Darwin's great insight to explain that "genius of a seed" that you extol without understanding. There are hundreds of billions of planets in the universe where nothing like the genius of a seed has come to fruition, for reasons easy to understand, not because a petty god (obsessed with being worshipped by small vulnerable things above anything else) decided those planets weren't worthy of his handiwork. You are blind indeed. The worst kind of blindness is lack of will to see. It's not that you don't know. It's that you don't want to know.

The word "point" in itself does not tell you what it is. A point on the real line: \( x \in \mathbb{R} \) A point on the real plane \( \left(x,y\right) \in \mathbb{R}^2 \) ... etc. Edited: A point is a locus, location, place in a set. When you say "point" normally you imply some kind of position (distance-->geometry, topology...). When you say "element" or "member" you normally imply just set theory. There's context missing. And as Ahmet suggested, "light cones", "faster than light", "hyperplanes"... That has nothing to do with your drawing or the concept of points. The impression I get is, again, you're trying to connect too much in one simple concept. Points don't need light in order to be defined.