It seems quite obvious now that Barry does not want to learn anything and is trying to push his own personal theories. Is there a way to request thread lock? :uhh:
Nope. Energy is constant for a given photon. You just wrote down the equation. Both h and f are constants (h is universal constant, f is constant for a given photon) so E must also be constant. Keep in mind that frequency tells you how many cycles are there per second and so it does not change...
It's not just a problem of applicability of equations. Have you tried solving Schrodinger equation for that problem? It's ridiculous compared to the simple classical solution. :yuck:
Thanks, now I understand where the 7% comes from. But I still don't get why negative curvature means finite universe. Can't infinite universe have negative curvature as well?
The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the...
I don't think it's meaningless. In introductory classes (where vectors aren't even defined properly) rows and columns are often just different notations for components of vectors.
If you're talking about simple real vectors (e.g. arrows in euclidian space) than the transposed vector is the same as the original. It's merely a matter of notation,whether you write the components in row or column.
In general case there is a distinction between row vectors and column vectors...
What I find most confusing about topology is that there seem to be many similar (or related) definitions and concepts and not enough examples for them. It helps a lot when I know an example that comes with definition because even when I forget parts of the definition I can always fill the blanks...
This is not true. You cannot do so in real numbers because ∞ is not defined on reals, but there exists the extended real number line, which defines +∞ and -∞. You CAN do some arithmetic operations on infinity in such case. It is commonly used when doing calculus, when computing limits for...
I don't think you'd have to generalize it. There is no general definition of inner product so definition for a given field is driven mainly by convenience.
The rule is that the sum must be finite. If the series does not converge than obviously you can't sum the two series, because you have infinity - infinity.
You can calculate with infinity if you calculate within the rules. The same holds for zero for example. You can calculate with it, but you...
Once again. It's irrelevant. It does not matter whether there are new rooms. There could be infinite number of new rooms in the hotel, but they are not needed to accommodate the new guest (as the paradox demonstrates). Let me try to restate the problem a little:
Hilbert's paradox of the Grand...
You're not getting the point that it's not important whether there is a room for the guest or not: you don't need it even if there is. You can fit the new guest into one of the old rooms and not use any new room. The premise that there is no room for the guest is not a premise of the mathematics...
##V^i## and ##S^i## are sets. I should have mention that before.
I see that with your definition of basis, but in the definition I used it wasn't obvious that the basis is maximal (linearly independent set that spans the space). It isn't even obvious that every basis of the vector space has...
No I don't. The dimensionality of space is undefined yet. I was hoping that this construction would allow me to define whether the space is finite or infinite dimensional.
I was thinking about axiom of choice for step 1 of the construct, where you have to pick an arbitrary vector from set, but...
Hi,
I've been trying to prove that every vector space has a basis.
So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of...