elfmotat

Because 12=1, and the rest of the elements in the row are zero. So [math]\sum_j (w_{ij})^2=1^2+0^2+0^2+...+0^2=1[/math]. EDIT: You're right, I read the question wrong. The simplest matrix I can think of which satisfies 1. and 2. is the identity matrix, but that doesn't satisfy the condition that n>>k.

Came across this interesting problem on the Feynman Lectures site. The answer is there, so no peeking!

Sure. A trivial example would be [math]w_{i1}=1[/math], and [math]w_{ij}=0[/math] for all [math]j\neq 1[/math].

Racist nonsense.

Sorry, but this is necessary:

Except a ribbon will curve toward your scissors, while wood will curve away from the knife. I imagine it has something to do with the way the natural fibers in the wood are agitated by the knife. Definitely a cool question.

Hi there Xblur. I'm not really sure what your question is. Are you trying to get a better understanding of inductance, or just the Henry unit?

No. It seems to be your trend that you think expansion should somehow be more powerful than other forces. It isn't.

In the realm where Newtonian physics applies, the effect of the Cosmological Constant is far too small to be measurable. I'm not sure what you mean. They both follow an inverse square law, but the electrostatic force between two electrons is of the order of 1042 times greater than the gravitational force. This is such a stupendously large number that gravity has a completely immeasurable effect for most fundamental interactions.

I don't know why you made four threads asking the same question, but spacetime curvature is completely consistent with experimental results. I'm not sure what you mean by "an observer observing from gravity free frame." I also don't know what it means to ask if something is "really" true. We know that if spacetime is curved in the way predicted by General Relativity, then this is consistent with observation.

By definition, if the speed of light is c in all inertial frames then there is no inertial frame where the speed of light is zero. Photons do not have rest frames by definition.

The tangential velocity is changing due to the centripetal acceleration. The acceleration is always perpendicular to velocity, which means that the speed of the object remains constant. If you add a nonzero tangential acceleration, the speed of the rotating object will increase.

[math]Q[/math] is the heat added to the system. This means that there's some heat transfer process, for example conduction, which is a complicated process that involves molecules bouncing around and smashing into each other. It depends on a number of factors, like temperature, the material, cross-sectional area, the thickness of the material, etc.

Work is path-independent if and only if the force acting on the particle is conservative. A conservative force is one which can be written as minus the gradient of a scalar potential function: [math]\mathbf{F}=-\nabla \phi[/math]. The work is path independent for conservative forces because the work done along an arbitrary curve C with endpoints A and B is given by: [math]W=\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}[/math] If [math]\mathbf{F}[/math] is conservative, then: [math]\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}=-\int_C \nabla \phi \cdot \mathrm{d}\mathbf{r}=\phi(A)-\phi(B)[/math] Gravity is conservative because it can be written as the gradient of the potential function [math]\phi=-\frac{Gm_1m_2}{r}[/math]. Magnetic forces, frictional forces, etc. are not conservative, so the work done in moving a particle through a magnetic field or when working against friction will depend on which path is taken.

I'm not exactly sure what you're asking. Are you trying to find the work done on particle 1 by [math]F[/math], so as to determine the energy of the system as a function of time?

Sorry about the reality check, but I wouldn't expect to get into either. The only people who get in to MIT or CalTech these days are people with absolutely perfect grades, tons of extracurriculars, and who have likely won science/engineering competitions or have published good research. Most schools also have an upper limit on the number of foreign students they're willing accept (no more than 10% of all admissions), so not being from the US will significantly hurt your chances as well.

It's important to note, however, that there are invariant quantities that every observer agrees on. Namely, proper time (the time a clock will actually tick off) and proper length (the length of something as measured in its own rest frame).

When taking a derivative of a sinusoidal function, if you're working in degrees then you pickup a factor of [math]\pi /180^{\circ}[/math]. This is because the derivative is "naturally" in terms of radians. So if [math]\theta[/math] is in terms of degrees, then the equivalent angle in radians is [math]\pi \theta/180^{\circ}[/math].

Did you read the part of my post where I said, "Now you're getting too small to be considering gravity... we've moved into the realm where Newtonian physics and gravity no longer apply"?

Massive objects can't travel at the speed of light. You can't use physics to explain scenarios which violate physics.

Now you're getting too small to be considering gravity, but if you plug in M=proton mass then the prediction is that gravity will become repulsive at ~1.1 feet away from the proton. But, like I said, now we've moved into the realm where Newtonian physics and gravity no longer apply.

Phase velocity is [math]v_{p}=E/p[/math], and group velocity is [math]v_{g}=\partial E/\partial p[/math]. For light, [math]E=pc[/math], so [math]v_p = v_g = c[/math].

Honestly, I just read textbooks. If I find the topic interesting then I have no problems maintaining focus. If I'm bored I might go back and work out a few practice problems just to solidify everything in my brain.

It works exactly the same way that the usual law of gravitation works. So it works for all macroscopic objects that aren't particularly dense, and which are moving slowly compared to light.

I don't understand what any of your pictures mean at all. And you still haven't defined [math]\Phi_{alt}[/math]. And I also don't see what any of this has to do with Nordstrom's theory. And I certainly don't see what any of this has to do with unification.