hobz

I have a few questions concerning the Faraday cage. As I understand it, the electric field inside a cage build from a conducting material, is zero. Does this mean that the sum of electric forces acting inside the cage is also zero, or are there simply no forces at work? The conducting material opposes outside electric fields, by aligning the electrons in such a way, that the electric field inside is zero. But what if the total number of electrons in the cage, are not "enough" to oppose the outside field? That is, if a very strong outside field is applied, how can a limited number of electrons align to oppose the field? And finally. The cage can be a grid, or bars. But is it also a Faraday cage to build a cube of conducting material, and stand inside the cube, or is there some minimum of what constitutes a cage. What determines such a minimum? Thanks!

Thanks for answering. You are right, he puts on a third filter. However, your explanation assumes wave nature?

Does anybody know a working definition of real? Is "real" that which can be measured? Is a thought real? Is a dream real? Is an electromagnetic field real? Are abstractions real?

I have been under the idea that experiments either shows the wave nature or particle nature of things, never both at the same time. I recently watched a shows called "The Mechanical Universe" in which a professor demonstrates an experiment with light and Polaroid. He explains the phenomenon in which no light passes through two cascaded polaroid lenses, polarized perpendicular to each other. He then turns the second lens at an angle of 45 degrees, and now some light passes through. He says: ".. It is very easy to understand so long as we believe that light is a wave. But remember, light is also a particle, and there must be a particle explanation of how this occurs as well." He explains the particle nature as follows. The first filter lets all vertical polarized photons through, and rejects all other. The probability is 50% of either event. The second filter only lets through horizontal polarized photons, of which there are none. When he then turns the second polaroid oblique to the first, he states that the photons thought they were either up or down, but now are either oblique along the way of the polaroid or perpendicular to that. What kind of explanation is that? "You thought you were"? How does this really work? It seems to me as though he claims that all phenomena have a wave explanation and a particle explanation, although the double slit experiment cannot be explained be either, but is explained as experiencing wave-particle behavior.

Can you guys recommend a book, a chapter from a book, a website or video, that explains this in greater detail?

They are real. Windows assigns CPU-time in slices of size tens of milliseconds. This accounts for the "preferred" intervals of 40, 50 and 60 ms. The intervals in between are, to the best of my knowledge, a random phenomenon occurring from the intrinsic CPU scheduling algorithms of Windows, along with all the other processes also competing for CPU-time at the same time (simulated by multithreading at least) as the thread I am running. I read somewhere that over time the CPU burst times (i.e. the time a process actually uses the CPU) are exponentially distributed, thus leading to some understanding of the results shown in the graph. Assuming that we displace this exponential distribution, which in fact we do by requesting CPU-time every 40 ms, then a double sided exponential distribution (Laplace distribution) would seem a good approximation. Of course, the Laplace does not account for the spikes at 50 and 60 ms.

Thanks for the insight! It would appear that in this case the maximum is peaked. This is a bit unfortunate, since the pdf with the most meaning assigned to it (prior to the experiment being performed) would be the Gaussian distribution.

So [math] \frac{\partial f(x(t))}{\partial t} = \frac{\partial x}{\partial t}\frac{\partial f}{\partial x}[/math] sums to one? Why?

I feel like providing you with a bit more information about my experiment. It would be nice with a small discussion. I am running a thread (in .NET) on a computer (running windows). Every time the thread is executed (every 40 ms), I timestamp the instance of the execution, leaving me with the time between each thread execution; my samples. Now, I have, because of the intrinsic structure of windows thread management and time slicing (distribution of CPU-time to each running process), assumed that these samples are normally distributed around 40 ms. The assumption is more or less based on CLT, because it more or less supports that unknown phenomena are probably normally distributed. Take a look at the histrograms of the samples. The first one has a resolution of 10 ms. At first glance this might be argued to look normally distributed. Now, the resolution is 1 ms. And 1/10 ms. These look more like a Laplacian distribution. However, I cannot verify that they indeed are Laplacian, and not logarithmic or some other distribution.

I have collected 53000 data samples, and I want to found out how they are distributed i.e. normal distribution, laplace distribution etc. Is there a general way of determining this?

Thanks for the replies! ajb, the fact you state at the beginning, is that derived from [math]\int f(t)g(t) dt[/math]?

Can someone explain to me why [math]\frac{dP(t)}{P(t)}=r \, dt[/math] becomes [math]\log |P(t)| = rt + C[/math] when integrating on both sides? It's the left side that is my problem. The right side is understood.

Your help is very much appreciated! Thank you. Out of interest, what is your profession?

Thanks for the code! Will try it when I get to a computer, that has python. I am looking into the DFP for optimization, and there the search direction is based on purely the information from the (negative) gradient. With [math]s_k = H_k \cdot (-\nabla f(x_k))[/math] where [math]s[/math] is the search direction and [math]H[/math] the hessian, how can I obtain the gradient at a single point? Just using your functions, and evaluating at a few points more (around x_k +/- delta)?

Nice. Think I'll stick to dfdx2 = (f(x1,x2+h)-f(x1,x2-h))/(2*h). Do you know of a code example of this?

I am trying to teach myself numerical analysis, and I have come to numerical differentiation. I understand how this works have only one independent variable. Now suppose I have functions of two or more variables. How can I calculate the partial derivatives and the gradient numerically?

Yeah ok, that was kinda stupid. Reused a function from a sampled function, didn't notice the 2*pi. Thanks for clearing it up!

I have written the following MATLAB code h = 10e-12; f = @(x)cos(2*pi*x); x = 0:.01:pi; fd = (f(x+h)-f(x-h))/(2*h); hold on plot(x,f(x)) plot(x,fd, 'r') The result is quite surprising. fd, which is the derivative of f evaluated numerically, has an amplitude roughly 6 times larger than what the real derivative is. Can anyone explain how this comes to happen?

Nice. Didn't know that. So how can the wave exist in the material, if the holes are bigger?

Mr Skeptic: So the wave is reflected due to an opposing wave induced in the metal? And this opposing wave can only be induced, if the metal is spread out enough (meaning that holes in the metal must not reduced the metal below a certain size), to allow a full wavelength of the incoming wave to induce a full wavelength of the outgoing (opposing) wave? If a hole disrupts a full wavelength, then the wave cannot be "created" in the metal, and thus not oppose anything? Can anyone point me to a direction, in which I can read more about this description of opposing fields?

I agree, but those fall under what I called pathological examples. Anyways, I think I got it now. The second article uses a second order Taylor expansion, to get curvature information as well, where as the first article only uses first order information. Thanks to all who replied. I appreciate it.

Thanks for your reply! I too sometimes claim that skimming is reading on wikipedia. So what article 1 states is that, the algorithm always will converge to a root (except in some pathological cases), however, it will only converge to the specific root x* if x_0 is close enough to that root, otherwise it WILL (again no pathology) converge towards another root? This fact could, in my opinion, had been illustrated better, if the illustration of wikipedia had two (or more) roots.

I find the following confusing. I compared the two Wikipedia articles with each other: http://en.wikipedia.org/wiki/Newton%27s_method and http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization The first one tells us that with [math]f'(x_{n}) = \frac{ \mathrm{rise} }{ \mathrm{run} } = \frac{ \mathrm{\Delta y} }{ \mathrm{\Delta x} } = \frac{ f( x_{n} ) - 0 }{ x_{n} - x_{n+1} } = \frac{0 - f(x_{n})}{(x_{n+1} - x_{n})}\,\![/math] then [math] x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\,\! [/math] No restrictions what so ever as to the choice of [math]x_0[/math] (the initial value). However, the second one, uses the Taylor expansion of [math]f(x)[/math] [math] \displaystyle f(x+\Delta x)=f(x)+f'(x)\Delta x+\frac 1 2 f'' (x) \Delta x^2 [/math] and then restricts [math]x_0[/math] to be chosen sufficiently close to [math]x^*[/math] to ensure convergence. Why this difference?

I see. Does a graphical interpretation of the wave even exist? And could it explain (graphically) why the wave is not able to penetrate the grid?

Is this because of the angular resolution and Rayleigh criterion? I am not sure if I understand what you mean by "size"?