Xerxes

This is obscene. Evidently you have no wheel-chair users in your family - I do, so f.u.c.k your smart-aleck comments.

Yeah? Then how about this I really doubt you understand the subtleties here. Get yourself a good text, and read about gauge theory - no, I don't understand it, if you must know (but you will, right? as you are cleverer that the rest of us). There you will learn about gauge invariance, and how it relates to Lorentz invariance.

Probably not, if the past is any guide for the future. Tell me, though; is this considered a serious topic among physicists? It would surprise me somewhat, although I am quite aware that judicious changing of signs in any equality preserves that equality. Browsing through A. Pais's excellent biography of Einstein (if you haven't read it, you really should), I found this quote from Planck: All equations of mechanics have the property that they admit of sign inversion in the temporal quantities. That is to say, theoretically perfectly mechanical processes can develop equally well forward and backward [in time] p.83 It's true he seems to have been making a rather different point, one that I believe Boltzmann's treatment (invention?) of the second law of thermodynamics addresses. I have a rather dim recollection from college that this employs just the sort of probabilistic arguments the quantum jockeys use - mm.. well no I see this is not quite right, see if I can dig out my notes (if anyone cares). But there - even the father of quantum theory believed in time travel!! (just kidding, I know he didn't - that was precisely the context of the quote I gave)

Ouch! And me saying it's part of my favourite identity? So sorry folks. Trivially, yes, of course. It's been a while since I looked at the internal structure of the Euler identity, but I suspect it makes use of the property which is (unaccountably) causing difficulty to some here. Like, assuming the result you want to prove?? Aaargh, Even worse - I said 2(-1) = 1!! Will I shoot myself now, or later? *Blush, blush* Sorry for any confusion

Well which is it chaps, make up your minds - rotation or reflection? For the record I contend it's neither, but that's by the way. Wow, j = -1? Why not just say so, then? You don't need imaginary units for that. Very pretty' date=' I agree, part of my favourite identity, if you must know. But how is it relevant? Surely you're not suggesting that, since [math']e^{\pi i} = 1[/math] and [math]e^{2\pi i} = -1[/math] this somehow "proves" that 2(-1) = 1?

Somebody help me here; is "elastodynamics" an accepted term? Although I'm no physicist, I've never heard it Of course you can't, as it is manifestly false; the "real world", as you choose to call it, is most definitely 4-dimensional. Specifically, it is a 4-manifold. Well, first, it's probably NOT knockout stuff, at least judging by your posts at http:http:////www.thescienceforum.com/index.php?sid=d9779770e7fc8a45f5b273aaebed7cfb. And second, you are an arrogantly tedious moron on any forum you choose to infect, who should learn to back his claims with something other than wishful thinking.

Ok.

Er, um... the context here is mathematics, isn't it? Or have I come to the wrong party? Then you are too self-effacing, my friend, I was talking directly to you. You made some assertions, specifically that equivalence was not well defined, with which I disagreed, what's wrong with that? You may disagree with my presentation, I'm cool with that, but why might you think it was misplaced? Sure I do, but it's not in the mathematician's lexicon, as far as I know.Is this a mathematics forum, or what? If it is, am I not allowed to give the mathematician's definition of terms, irrespective of what others might say here?

"Semantics"?? So if I tell you the thing I drive to work in each morning is a "table" and the thing I take for walks, goes woof and chases rabbits is a "telephone", that's just semantics? You may think so, I don't; it's a matter of agreeing definitions, which is what a lot of the language of mathematics is about. I gave the agreed definition of equivalence - notice the word definition I can do no more - I gave the precise definition of equivalence. What does "analogous" or, god help us, "or whatever" mean? Give us definitions. I have no idea what this means. Do you?

The claim is that there's no precise definition of equivalence. I claim otherwise: A member of an arbitrary partition of any space is called an "equivalence class" iff, with respect to the relation ~ the following conditions are met: x~x (reflexivity) if x~y, then y~x (symmetry) if x~y and y~z, then x~z (transitivity) for all x, y and z in [x]. Under these circumstances the relation ~ is called "an equivalence relation". Merely because the partition is arbitrary doesn't mean there can be no precise definition of when that partition induces an equivalence relation. If by 3d you mean 3-dimensional, you are wrong. The sphere (note the definite article here) is defined to be a 2-dimensional object. You may call it the 2-sphere if you want (but most people don't), just as the 0-sphere is a line, the 1-sphere is a circle, the 3-sphere is a ball etc. Anyway, who said we were doing topology, I never saw that stipulated.

This is, of course, nonsense. A circle is a 1-sphere, as I told you a few days ago and this is worse. You really believe this? There most certainly is, where did you get that idea from? Why? Local coordinates work just fine. Why do you think you "require" coordinates on the ambient space for a 3-manifold, but not for a 2-manifold? What, for example, do you think the ambient coordinates might be for the 4-manifold? (which I can assure you does exist - it's called spacetime, by the way)

Forgive my uninvited bolding, but I think this is great advice. For my sins, I subscribe to another couple of fora, where I try to "teach" what I have recently leaned. It really does test your understanding, when neophytes ask seemingly simple, but actually penetrating questions, likewise when experts challenge you for proof of your assertions. Don't get me wrong, I am an experienced teacher at grad level (not in physics or math, though). But when students come to me, as they frequently do, begging for a place on a PhD programme, I always ask the same question: do you really believe, deep down, that you will be in the top 10th percentile of all (mostly worthless) PhD.s given out these days? Otherwise, forget it; you'll end up being a PhD car-parking attendant, or worse, a lab-technician

I just saw this edit. I must be truly dim, but consider the following. You will agree that each p, q in P is integer, right? So let (p,q) be an element in P × P with [math]+_P:P\times P\rightarrow \mathbb{Z}[/math]. (Actually you wrote [math]+_P:P\times P\rightarrow P[/math] which is probably a typo, no?). Let [math]+_P(p,q) = x \in \mathbb{Z}[/math]. Now consider [math]+_{\mathbb{Z}}:\mathbb{Z}\times\mathbb{Z} \rightarrow \mathbb{Z}[/math]. If p and q are integer, then (p, q) is in Z × Z with [math]+_{\mathbb{Z}}(p,q) = x [/math] again. Being bold I'll say that, if two operations give the same output for the same input, then they must coincide. Am I mad?

Umm. So, if P is "just a set", and set theory knows nothing of +, how can you then assert that ? Recall I suggested the notion of an "algebraic set", i.e one with a binary operation. Oh? 3 +5 is undefined? Surely not. I completely agree, but not for the reasons you gave. Binary operations, one might say, are insensitive to the space they operate on. True, startling as it may appear. As you did; +: X × X → Y, ×: Y × Y → Z, where possibly X = Y, Y = Z, I'm reasonably sure all other imaginable operations can be derived from these.

Umm, well, I still don't see your point. Every group G has at least one subgroup, that is G itself, by the definition. How does this comment work here? But this is a property of P, not the operation +. Sorry to sound assertive, but; +: Z × Z → Z defines an abelian group (let's say) whereas +: P × P -/-> P, rather +: P × P → Z, hence P is not closed under +, and therefore P is not a group, therefore not a subgroup of Z. But for certain sure, P is a proper algebraic subset of the algebraic set Z. Are you suggesting there are two "varieties" of +? PS (by edit): In case there may be hard feeling here, I am merely suggesting that what you called your "smart-ass" question was well motivated, and, unless your prof could give a better explanation, (s)he was wrong to "roll their eyes", I think.

Tom, I can't convince myself of the truth of this. So, let's call a set with a binary operation an algebraic set. Grant me that, if A is an algebraic set, then any subset B of A "inherits" the operation from A. Now consider the algebraic set Z; this is an abelian group under the operation +,by the closure axiom. The subset P of Z, the primes, also admits of the same operation +, but P is most definitely not closed under + and is therefore not a group. You surely must have closure as an additional axiom? I cannot see in what sense the + on Z is said to be a "complete" operation, whereas the + on P (it's subset) is partial. Did your prof give any more info?

Oh? In what sense, please say. Then damn and double damn, I wasted half my life on this failed enterprise called science. Well, we agree on this at least. (Prat)

Ah no, don't belittle yourself here, you did fine. But, as you admit, you did it with a physicist's skim on it, and also while drunkenly skinning a rabbit! Cool! I didn't understand a word of that, sorry. Do I need to? I doubt it, I haven't even read the Boy's Own Illustrated Guide to String Theory. Fine. I find it sad, though, it is an interesting subject in its own right, not that that I am formally trained or anything. Yeah, someone else suggested I look at that, it's way too advanced for me.

No shit? I have trouble explaining it to myself, and still I'm not sure I get it. Ah well. But oddly enough, this is a subject I am now ploughing (plowing?) through, alone. I have Fulton & Harris, and I know a bit of Lie Theory, but I'm finding it really, really tough. Maybe you'd like to start a pedagogical thread on reps?

First this: Ben: I concede the tone in my last was somewhat haughty. I apologize, I hope you're not terminally miffed. Now, this is less trivial than it might at first appear. OK, the answer to your question is not "that's just the way it is", nor is it "it's a mathematical abstraction". Rather it is this: an object is said to be one-dimensional if the notions of width and length have no meaning. Let me try and explain. First we need to distinguish at least two different "sorts" of dimension. Consider a circle. You need a sheet of paper on which to draw it, right?; this sheet is a 2-dimensional object, or "space" as you might say; the circle "lives" in a 2-space. Likewise, a sphere lives in 3-space. Is there a rule? Let's call the space an object lives in it's ambient space. Let's further say that any object in this space can be uniquely located by reference to a coordinate set whose cardinality is defined to be the dimension of the space. Let's take this as the definition of the dimension of an ambient space. Fine so far, I trust. Let's now consider the objects in some n-space, and apply the same definition of dimension. How many "numbers", or better, parameters, do I need to uniquely specify a point on a line? A circle? One in both cases, obviously; these are 1-dimensional objects. How about a sphere? Two, latitude and longitude; this is 2-dimensional. And so on. (you can call these parameters coordinates if you want, like Ben did, I'm not sure it's standard, though). The pattern seems to be that the dimension of an ambient space is always greater than that of its embedded objects. There are theorems out there to this effect, you don't need to worry about them, though. The reason I rambled on about ambient spaces is this: the equation that describes the unit circle, x2 + y2 = 1, for example, is a function on the ambient space, not on the object itself. Hence my opening comment: it makes no more sense to talk about the length and width of a 1-dimensional object, zero or otherwise, than it does to talk about north of the North Pole; it doesn't parse. And finally, in this overlong post. (This is the sort of thing that causes my mates throw beer at me): the circle is "properly" referred to as the 1-sphere, the sphere as the 2-sphere, the ball (aka "solid" sphere) as the 3-sphere, etc.

Ben, although I agree with most of what you say, I find this a little over the top, in fact rather offensive. That raises issues a so-called science forum is not qualified to deal with. So leave it aside. But.... you wanted a response to your first post on the original question. Obviously I don't know, not being a physicist, but I will say this: You start by saying You mean, of course, a set of coordinates. And next: Two problems here: although I am anticipating slightly, the topological property you refer to is not "isomorphism" but "homeomorphism". I can explain if you want. Moreover, topological spaces can be homeomorphic regardless of what you call "kinks and cusps"; as you rightly say, the only criterion is the number of "holes" they have (this is the connectedness property, well there's more to it than that), kinks and cusps are OK. Neither can I convince myself of the truth of this Topology comes in two flavours - point-set topology and algebraic topology. Neither of these has anything to do with surfaces. Maybe you're thinking of manifolds here? And in any case, the word "topology" is widely abused; it is, technically, a set T of subsets of some set S which has certain defined properties, one of which posits the existence of the topological space {S,T}, on the other hand, "topology is used to refer to the study of such spaces. And I thought mathematicians liked precision! Oh, and finally, you say As the OP was about dimensions, you might have been wise to specify the 3-sphere; unless otherwise specified, the sphere is taken to be 2-dimensional.

Make that three.

I just noticed this thread. I really enjoyed contributing to WiSci (where I was ben, by the way), and would love to see it "re-born". Here's what I think should be done. Delete all, but all, imported articles, and give contributors free reign. Let all visitors contribute. If registration is deemed desirable, let it be really easy: register then edit. I am quite willing to re-enter the "fray", but you get to feel sort of foolish talking to youself. Anyone else out there willing to give it another go? It's such a good idea. And I still have a bit of mileage left in me.

For "not too far" read "equal to". What does "require speciation" mean? The definition of evolution? Of course it doesn't, otherwise evolution would be discontinuous. What do you mean by "below" here? And "shuffling" is most decidedly not what you should say. Do you mean recombination? We are talking mutation here, not recombination (although it can be reasonably argued that this itself is a factor in the evolutionary process) You might. And even if you did, you would have to argue back, using the continuity concept, that the differences between members of population A, over time, are part of the continuum which leads to the emergence, from A, of population B, arbitrarily defined as a separate species. These differences are fully accounted for by change in allele frequency over time. C'est tout.

Well, it is the accepted definition of evolution. If change in allele frequency is a "minimal definition", you might first explain what this term means (for myself I have no idea), and second, if it does have meaning what the "non-minimal" (= maximal??) definition is.