joigus

Linearity is one thing. That's inherited from linearity of the derivative operator. Closure is another thing. That's what Studiot and Taeto are talking about, I think. Because ex, exsin x, excos(x), sin(x), cos(x) are meromorphic (analytic in C ==> analytic in all R) I see no problem with the domains. Closure can be assured by inspection. The most involved cases are exsin(x), excos(x). But, \[\frac{d}{dx}e^{x}\sin x=e^{x}\sin x+e^{x}\cos x\in\textrm{span}\left(A\right)\] \[\frac{d}{dx}e^{x}\cos x=e^{x}\cos x-e^{x}\sin x\in\textrm{span}\left(A\right)\] That would be my answer.

I like the way you say "perisheth," Rippa.

Thanks for the tip. I'm here both to learn and test my knowledge and understanding. Language itself is one motivation too.

OK, it's been ages since you posted this, but I couldn't resist. I'm just refreshing my linear algebra. Then I can pick a basis in which, \[A=\left(\begin{array}{ccccc} a_{1} & 0 & \cdots & 0 & 0\\ 0 & a_{2} & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & a_{n-1} & 0\\ 0 & 0 & \cdots & 0 & 0 \end{array}\right)\] And, without loss of generality, \[v_{0}=\left(\begin{array}{c} 0\\ 0\\ \vdots\\ 0\\ 1 \end{array}\right)\] Generic n-vector: \[x=\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n-1}\\ x_{n} \end{array}\right)\] Eqs. render as, \[a_{1}x_{1}=0\] \[a_{2}x_{2}=0\] \[\vdots\] \[a_{n-1}x_{n-1}=0\] \[0x_{n}=1\] So no solution for xn.

Yes, exactly. I didn't want to get involved, but I thought it could be useful. False friends are a minefield.

We are all laypeople in some sense or other, aren't we? You're most welcome.

You don't have to apologise. I understand perfectly what you mean, and I was just trying to apply an old Chinese technique which is called "koan" for those kind of difficult questions about "I." It's a well-meaning technique. It's about making you drop our common human need to stick to the "I." In more scientific terms, an electron in my brain is fundamentally indistinguishable from an electron in yours, or another one being kicked off from an atom in the atmosphere. Quantum field theory tells us that elementary particles are just instantiations of one thing called the quantum field. Information is the relevant quantity for describing an "I," or any other physical object. Very recently a very good friend of mine has died. He was younger than me. I lost my parents when I was very young too. To me, all those people are still living in the only sense that I can find physically meaningful: They uploaded software snippets and applets to my brain, so they are still in the world in this particular sense of information processes. Some day I will die too. Hopefully, I will be able to upload my applets --those that prove to be useful, or good in any sense--, to somebody else's brain. That's the only way I can conceive of in which we can perpetuate ourselves. I don't mean to be facetious; only to bring some consolation to you by trying to make you feel more relaxed about the eventual loss of the "I," but I can't think of a better way to finish except with another koan: What is it that makes you John? I hope that helps.

Your English is next to perfect. The only thing I've been able to spot is "I have a doubt." I would have phrased it as "I have a question." More idiomatic. I'm bilingual Spanish-English and have taught both, so if you ever have any nuance, false-friend, etc.-related question, be my guest. The cogujada is a beautiful bird, by the way. Good nickname. Except typos, of course.

Beautiful, and tantalizing. Interesting question. My guess would be no, and that rotation of the accretion disks is mostly driven by local clustering in the accretion area of the newborn star.

Interesting... Tidal effects come to mind at your suggestion, because those are second-order effects, which requires an order-2 tensor.

In other words, ask yourself, is it true that, \[\left[R_{\pi/3}\right]_{\mathcal{B}}\left(\lambda\boldsymbol{u}+\mu\boldsymbol{v}\right)=\lambda\left[R_{\pi/3}\right]_{\mathcal{B}}\boldsymbol{u}+\mu\left[R_{\pi/3}\right]_{\mathcal{B}}\boldsymbol{v}\] for any u, v, lambda and mu?

Here's a starter: Horizons problem (Spatial) flatness problem Cosmological constant problem (vacuum energy) Exact solutions of GR alone could not account for the first two. Part of the reasons why cosmologists moved to inflationary models. More in general (don't forget we're living in the fine-tuning era of physics and cosmology) and how to get over it: Fine tuning Also, and on a different order of questions, but related, what are your predictions for these?: \[\varOmega_{\textrm{matter}}\] \[\varOmega_{\varLambda}\] \[\varOmega_{\textrm{dark matter}}\] \[\varOmega_{\textrm{radiation}}\] all of which can be assessed from observational input.

I agree in a short-winded kind of way.

It depends on your "pervective." 👍

Boy, was that a good explanation! +1 And this was intended as a joke. Coordinates mean nothing, it's the metric tensor contracted with the coordinates, as Markus so brilliantly has explained. Just to clarify...

I'd say a couple of things present themselves as very fundamental and intrinsically quantum (non-classical.) 1) Finiteness of the quantum of action (quantities such as energy x time, momentum x displacement, angular momentum) 2) The need to express probabilities in terms of a more primitive quantity called the probability amplitude, which is complex. But it's impossible to explain quantum mechanics in a few sentences and without mathematics.

Yes! The "it's complicated" one would have been my choice. Thank you for the article. I haven't been able to read it yet, though. Octonions are very promising. John Baez has a lot of stuff on octonions too. They relate topologically to the the spinors. They kind of split into spinors through a mapping that renders the algebra non-associative, as octonions are non-assoc. Thank you for the references. This topic is very interesting.

I meant to +1 Strange on the update to Couder. I remember back when I studied quantum mechanics, at some point we tackled the really hairy aspects of the mathematics. There were several attempts to formulate QM based on real numbers, quaternions and even octonions, if I remember correctly. The argument was that using a complex-number based mapping of the amplitudes seemed to be minimally essential to describing its properties satisfactorily. If that's true (I didn't completely understand the reason because it was not explained in detail,) it would be too much to expect that waves that are well-modeled by real functions would be able to reproduce all of QM. Especially fermions. Analogically mimicking photons would be a problem too, for obvious reasons. I'm confident that once we understand where the logical necessity of using complex numbers arises, we will understand the nature of a double solution better than De Broglie-Bohm. That's why I've voted for the 1st option. +1. Thank you. But those are pre-2015. Aren't they?

I know the answer to that one. Every time a newborn appears in this world after one's death, some form of auto-conscience appears again at some point in the future. The question is: Is that good enough for this form of auto-conscience that's asking the question?

That's only because I'm a remote object, and you're giving me the wrong coordinates. Apology accepted.

Boy, that's beautiful!

I meant "with weight equalling buoyancy." Yes, I have to teach chemistry sometimes, so have must think about those things. Although my chemistry laboratory practice seems very remote now.

Don't put words in my keyboard. The only one who believes in remote dilation as a frame-independent phenomenon here is you! That remote dilation is only in your mind. You don't even understand that, which says all about you as a "thinker." The fact of whether something suffers frame-dependent dilation or not does involve having things other than photons in it. In fact, in order to write down the geodesic equation for photons you must do an affine transformation, because their proper time is identically zero, so no common-sense clocking will help you describe their histories. But what am I telling you; you know next to nothing about relativity, that not being the worst. The worst being that you don't bother to examine your own assumptions, or anybody's criticism. Can you imagine Einstein telling Hilbert "please consider my silly mistake as a valid assumption"? Einstein quickly re-wrote his paper. Learnt much from Hilbert, and went on to publish one of the most important papers in the history of physics. I will pop up every now and then to see what experts and other serious thinkers have to say. Your post is only valuable in that sense. The only trouble is I will have to check for you twisting everything I or anybody else has said, just because you don't understand the first thing about relativity, you don't read what you write, let alone others, and you stubbornly stick to a bunch of silly pseudo-scientific propositions to no end.

Yes, I know, that's why I said "ditto." But then I realised there are even more reasons for confusion for physicists, so I just tried to throw them in, but there's always a risk that you're bringing more confusion. Totally, yes. We don't really know who is asking, though. Whether my explanation or yours have been helpful, we don't really know. I hope both have. I kind of have become a specialist in many common sources of confusion for the students. Believe it or not, sometimes it's in the language. Very recently I remember a problem on hydrostatics, and the source of the confusion came from how the statement had been worded. It said that the body was "floating," and it really meant that it had sunk in equilibrium with equilibrium equalling buoyancy. I don't know what word the person who wrote the problem would have used for floating like an iceberg. Maybe it would have been quasi-floating. That's a problem too. Oh, yes. You're right. I forgot. To me that would be negative. I also think considering it negative is quite standard, also in molecular biology. But it's not impossible that somebody somewhere could have a different criterion elsewhere. Is it different in any context that you know of in chemistry? It may well be. You're very interdisciplinary, and encyclopedic.

It is entirely possible that I'm overthinking it, in which case I apologize. But the way I see it, distinctions such as "work done on a system" and "work done by a system" are often confusing, especially when one is dealing with reversible work --in the physicist's sense--. Reversible work is of theoretical interest. And in order to define, e.g., variable pressure for reversible processes, you must make external pressure equilibrate internal pressure, so that the concept of "who" is doing the work becomes more confusing. I may have misinterpreted him, but Swansont's point that, Is of some importance here, I think. "Define your system," are key words for me in it. It's very easy to get lost in the different steps at which your definition might alter your formulae. For example, suppose somebody defined the different intensive thermodynamic parameters in different ways, while at the same time defined the criterion differently as to "who is doing the work." Example (homogeneous system with variable number of molecules): \[dU=\delta Q-PdV+\mu dN\] Maybe some "crazy author" decides to go one step further in an orgy of definitions and set the chemical potential as the negative of the usual one. One possible choice is (I think it's quite standard), \[P=-\left(\frac{\partial U}{\partial V}\right)_{S,N}\] \[\mu=\left(\frac{\partial U}{\partial N}\right)_{V,S}\] \[T=\left(\frac{\partial U}{\partial S}\right)_{V,N}\] But there could be others. The fact that pressure has a negative sign is just convention. No real meaning attached to that, I think. It's just a thermodynamic potential. And the real question is well away from any ambiguity when you express, \[dU=\left(\frac{\partial U}{\partial S}\right)_{V,N}dS+\left(\frac{\partial U}{\partial V}\right)_{S,N}dV+\left(\frac{\partial U}{\partial N}\right)_{V,S}dN\] Now, the signs there in the last eq. do not depend on any criterion. And I do confess I'm something of an unbearable stickler for sound definitions, axioms and language. Actually, I think the "usual" one is the opposite. That's kind of what I mean.