hobz

I a spectrum employing this technique, and I didn't know why it was used. Have you ever seen it?

What are the advantages of normalizing a DFT with the frequency? E.g. A plot of DFT*frequency as a function of frequency.

That's pretty interesting. For if the creation of the photon happens to occur at some point, then the photon is, as such, unaware of the frequency of the jiggling at the point of its creation but still manages to have the "right" frequency property. Steevy: With one electron, there should be no leaping.

Hmm.. so any acceleration will give off photons. Suppose I jiggle one electron at a certain frequency over a certain distance. Will the number of emitted photons be related to the distance, and how? How is the frequency of my jiggling related to the frequency (or wavelength) of the photon?

Suppose that I have one electron which I accelerate. How far a distance must the electron move in order to cause the emission of one photon? I.e. if the energy required to move the electron along a path must match the minimum energy of the emitted photon before the photon can be emitted? Sort of like an energy-meter (a measuring device for energy) that has to be filled before a photon is emitted.

Suppose I have an antenna where I drive the electrons back and forth using an alternating voltage source. The acceleration will cause photons to emit. Do they spontaneously appear? As photons carry momentum the accelerated charge will feel a force when the photon is created, right? Will this force accelerate the charge so that it will emit another photon? Can the photon explanation be related to the wave explanation directly? As the electric(and magnetic) wave created by the acceleration does not propagate in the direction of acceleration, I assume that the wave function is zero in that direction.

How and when are photons created?

So is the tensor what takes a description in one coordinate system to the description in another? Or is it the description it self?

According to this, "The upper indices are not exponents, but instead different axes. Thus, for example, x^2 should be read as "x-two", not "x squared", and corresponds to the traditional y-axis." What is meant by "traditional y-axis"? Could you point out a simple example of this?

This is the most lucid explanation I have come across. I was really looking into tensors, but please do recommend books for fluid introduction as well. Thank you. Perhaps, but then again, I bet tensors (and their related background) are not taught as single-post forum replies I am having trouble with your notation (or the summation notation). What does it mean? Also," Tensors are first order geometric objects that form a linear representation", what does it mean to form a linear representation?

What is a tensor and why is it useful? I have grabbed "Vectors, Tensors and the Basic Equations of Fluid Mechanics" by Rutherford Aris, but it is not a gentle introduction (some of the notation used is not explained at all!).

You can compare the piezoelectric material to a capacitor where, instead of charging it with a voltage, it is charged by mechanical stress. So applying stress to the capacitor creates a voltage across the material. If you were to use this voltage (by allowing charge to flow) the voltage would drop until equilibrium was reached. Removing the rock would re-create a potential difference although reversing the sign. As to how it produces the voltage, I suspect that there would have to be some electric dipole moment of the material that is altered by the application of stress.

To my understanding, the wave-nature of energy can explain a lot of phenomena with the exception of interaction with certain other energy manifestations e.g. light interacting with an electron. Is the reason that the wave-nature cannot explain this that the superposition principle of a wave simply will add the two wave-forms of the photon and electron, and not cause a change in either's momentum? If so, could the wave-theory be modified to somehow exclude superposition under some circumstances?

How does a digital input work? I suspect that it's a flip-flop with a clock input, so that the value of the digital input is sampled at the rising (or falling) edge of the clock. I see a problem with this though. If I wanted the digital input to be a counter, then, if the input is another clock running faster than the system (sampling) clock, I would not be able to count accurately. So how are digital inputs usually implemented?

Ah yes, and then L'Hopitâl doesn't apply since the limit does not converge to 0/0.

Interesting. I will try and follow your suggestion, when I get some free time. I've used [math] e = \lim_{x\rightarrow 0}\left(1+n\right)^{\frac{1}{n}} [/math] and substituted [math]a[/math] with that definition of [math]e[/math] in [math] \frac{\mathrm{d}}{\mathrm{d}x}\,a^x = a^x \cdot \lim_{h\rightarrow 0}{\frac{a^h-1}{h}} [/math] which eventually results in [math] \frac{\mathrm{d}}{\mathrm{d}x}\,e^x = e^x \cdot \lim_{h\rightarrow 0}{\frac{h}{h}} [/math] Edit: Btw. Does the limit of [math]\lim_{h\rightarrow 0}{\frac{h}{h}}[/math] exist? Applying L'Hopitâl seems to yield 0/0 which is undefined.

In your derivation you use [math]f'(n)= a^n(\ln(a))[/math], but that is the very thing I am trying to prove. So how can the proof contain the original problem?

Interesting. Logarithms are far older than Riemann sums, as far as I know. Although the ancient Greeks are generally credited for alot of things, I doubt that javascript is one of them.

Thanks!

What is the limit of this thing [math] \lim_{n\rightarrow 0}\frac{a^n-1}{n} [/math] It has come up in an attempt of mine to find the derivative of [math]a^x[/math]. According to my derivatives table, the derivative of [math]a^x[/math] should be [math]a^x \ln(a)[/math], suggesting that the limit should converge to [math]\ln(a)[/math]. The question is, how do I arrive at this?

The opposite could result in overflow, if y and z are sufficiently far apart, and x is large enough. I wonder if these rules of thumb are summarized somewhere?

So the point is, if you can come up with an approximation that is accurate down to the precision of the computer, then it would be worth examining whether one calculation method would perform better than another.

I see! So the implicit requirement for a function to be smooth in order to be differentiable, can be used "in reverse" to find out if a function is smooth. This also implies that all smooth functions can be represented as a Taylor series?

I agree, but I think you misunderstood my question. I am asking how these things were calculated. Not how the need for them (or their names) came to be. Merged post follows: Consecutive posts merged(Sorry for the "How was", but I appended logarithms at the last moment, and forgot to change to "were")

I was wondering how the ancient Greeks calculated sine, cosine, etc. My guess is that they measured, and put the values in a table. Likewise, how was the first logarithms (or inverse functions in general) calculated?