DQW

If the set of positive real numbers is a "ray", then it is an infinite set. The set of all (non-negative and negative) real numbers is a "line", and that too is an infinite set, and is of the same "size" as the first set.

Yes, it would take 4-d atoms. But my question is this : If you can concieve of a 4-dimensional object, why should you have trouble conceiving of a 4-dimensional atom (which itself, is nothing but an object) ? All (non-fundamental) physical objects (from atoms to elephants) exhibit the same dimensionality as the space they live in.

I have learned that this is not a great way to judge rigor. Detail or comprehensiveness perhaps, but not rigor.

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But this isn't true ! Experiments clearly show that the electrostatic interactions between electrons in a current carrying conductor are a very small perturbation that can very well be neglected in describing the microscopic dynamics of conduction. It is essentially for this reason that the almost naive Drude Theory does spectacularly well in predicting the electronic properties of the alkali metals. In fact, models all the way up to the Tight Binding Model (which is resoundingly succesful in predicting band structures of various materials) are based upon the "independent electron" assumption - that basically, electron-electron interactions can be neglected. It is only with the Hubbard model (and its variants in the Anderson-Hubbard, Mott-Hubbard, or the Bose-Hubbard model) that you first start to deal with electron-electron interactions. Such models have known to produce new revelations only in the more exotic materials like Mott insulators, but even they are extremely rudimentary in the extent to which they model many-body effects. The difficulty of dealing with electron-electron interactions was next lowered by the approach utilized by the Fermi Liquid Theory (where weak interactions are renormalized into the effective mass of a quasiparticle - which is now, essentially a non-interacting particle). Strongly correlated electronic systems (BECs, Superconductors, etc.) are found at the top of the "exotoc materials" ladder, and it is only in these sytems that you need a robust framework capable of modeling the interactions between the electrons. For regular conduction in ordinary metals though, electron-electron interactions play no role.

I hope you did not infer this from Dave's posts - because he never said anything about "mobile dimensions". He was talking about the mathematical concept of dimensions as the basis vectors of a vector space. This is an algebraic concept. It appears though, that you are interested in spatial dimensions, which is a topological (or geometric) concept. Simply put, the dimensionality of an object (or space) is simply the number of co-ordinates (or scalars) that need to be specified to identify every point in that object. For instance : A line, (or a circle or a parabola) is a 1-d object because every point on it is uniquely identified by a single co-ordinate. For a line (or a parabola of the form y=ax^2), specifying the x-coordinate of a point on the line (or parabola) completely specifies the location of the point. On a circle, specifying the angle (theta, from some fixed direction) completely identifies the point. A plane, or the surface of a sphere (the surface of the earth, say) is a 2-d object. Any point on this object can be specified using 2 co-ordinates. On a plane, the x and y co-ordinates will do, and on a spherical surface the [imath]\theta,\phi [/imath] co-ordinates. A sphere itself is a 3-d object as you need also specify the radial position to determine the location of a point in a sphere. Keep in mind that these are all mathematical objects. Physically though, all objects have the same (not lower) dimensionality as the space they inhabit. In our universe, this is (to an excellent approximation, at the very least) 3. It is also important, to specify an additional dimension (in the framework of classical physics) that is important to fully describing the location of an event. This dimension is time. It is simply a useful dimension to have in your metric if you want to describe dynamics. Note, however, that some physical frameworks (such as Quantum Mechanics) do not treat time as a dimension, but still do a great job of describing dynamics.

What is the difficulty with a 4D atom. If you can be content with 4D space, you should have no problem with a 4D atom (or for that matter, an n-dimensional atom, for any n)

http://mathworld.wolfram.com/Hypersphere.html

It looks like it chooses not to plot when z gets very close to 0. I can't see why this happens, but then, I know precious little about numerical algorithms. In fact, the only one I'm familiar with is Newton-Raphson, and I don't see that having any problems, but I have absolutely no idea what real graphing softwares use now.

In the rest frame of the ships, no work is done in both cases. In the rest frame of the water, the power consumed is 1000 HP in both cases.

This has been proven incorrect by counter-example : the Law of Cosines, for instance, should do. Things that are believed true (and can not be proved) are usually called 'axioms' in math and 'postulates' in physics.

I have no idea how graphcalc works or how it calculates square roots. I'm not sure if there's a "division by zero" hazard it tries to avoid. Have you tried moving the center away from the origin (or the x-y plane) ? Can't make a strong case for why that should work, but it's not too much more effort, so may be worth a shot, until someone comes up with a real fix.

Hmm...can't see too clearly. Do you have a bigger version of that ? It looks like you have most of the 2-sphere except for a "ring" near the x-y plane. Or is that the y-z plane ?

Does a "lay person" know the math of vector spaces, matrices, and (scalar and vector) calculus ?

M is the pole strength (proportional to the magnetic moment) - it is not the field at some distance from the magnet. Think of the electrostatic analogue, the force between two charges. [math]F = \frac {kQ_1Q_2}{r^2} [/math] [math]E_{Q_1}® = \frac {kQ_1}{r^2} [/math] [math]\implies F(on~Q_2) = Q_2 \cdot E_{Q_1}|_{r=r_{Q_2} [/math]

See above.

YT : pogo is right. Sine waves and circles are only 1d spaces. You can specify any location on a specific sine wave or circle using just one number. To describe a sine wave or a circle, however, you must have it live in a 2 (or more) dimensional space.

Thanks for teaching me how to read. I'll keep that trick in mind. The length of my big toe is related to mathematics, but talking about my big toe does nothing to answer a question about defining "infinity". As relevant as the fact that I can not in general, represent the lengths of my toes (in some standard length unit) using a finite decimal representation. And apparently, not a mathematician either (unlike some other members here). I'll keep that in mind, the next time I ask for help.

I may agree with the first two sentences. I still do not see the usefulness of "significance" - which I know only in the context of statistical deviations - in explaining infinite sets or points at infinity. And please do not forget that it was in neither of these contexts that you used the word. Your 'significance' had to do with line of sight or dispresion (at least in one specific example). While 'green' can be the color of an apple or the color of a leaf, it is still the same thing. What is the point of this statement ? You are merelysaying that an infinite series can converge. Okay, what's new ? You've already spoken for the usefulness of infinite sequences and ideas like convergence. It is almost impossible to separate mathematics from physics unless we are talking about very specific numerical relations. Au contraire, mathematics exists ABSOLUTELY independently of physics, and I can't see that this independence is dependent upon "very specific numerical relations". I shouldn't be speaking for matt, but I don't recall him saying any such thing. Please quote where he says this. And how does their being mathematical theories have anything to do with answering the OP's question ? If a post requires added verbosity to reveal a relevance to the topic of discussion, then it is (at least in the absense of this verbosity) off topic. If someone asks me for the equation of a straight line in cartesian co-ordinates, and in response, I talk about Minkowski, light cones, railway tracks, regression analysis, non-Euclidean geometries, Lyapunov exponents, waveguides and the least action principle, without ever writing down the needed equation, what have I achieved (besides telling someone that I know about all these things) ?

PS : Notice that the equation of the 0-sphere in cartesian co-ordinates can be written : [math] (x-c)^2=r^2 [/math]

The equation in cartesian co-ordinates is the natural extension : [math]\sum_{i=1}^{n+1} (x_i-c_i)^2=r^2[/math] In general, I believe the locus given by |x-c|=r holds for all dimensions, where now, x and c are elements of [imath] \mathbb {R}^{n+1} [/imath] and r is in [imath] \mathbb {R}[/imath].

This may just be me, but I dont see how either of those links is significant !

c: center; r: radius

The 0-sphere, I believe is the locus described below : {x : |x-c|=r; x,r,c in |R } This describes the pair of points c-r and c+r, where c,r are in |R. PS : forgive the sloppy formatting - I'm too sleepy now.