Using Fourier to get the numerical FT of a Gaussian
What do the X and Y axis stand for in the Fourier transform domain?
How can I start learning about political philosophy? I'm interested in learning about political philosophy. What are the best ways to start? Are there any websites for beginners? Are there any books such as the "For Dummies" series? Are there any audio or video introductions to familiarize myself with the terminology? Oddly wide spaces in subsection title I have a problem, this is what it looks like:     And this is the code:  \subsection{Bakterienst\"amme und Anzuchtbedingungen}   Where does the huge whitespace come from? I'm trying to set an alias for sudo !! in Bash. I tried alias sbb='sudo !!', but it interprets that as a literal  !! and prints   sudo: !!: command not found   If I use double quotes, it substitutes the double bang in the string itself, so that doesn't work.  Is there any way to make this work? Or an alternative alias? ` How can I stand on the ground? EM or/and Pauli? Does a conducting wire give off measurable radiation? I am trying to find a story where a children's nursery turns into a jungle I read this short story(I believe it was under novella length, but I could be mistaken) a while back, and I am trying to identify the title for a friend.   Plot:      Parents(male and female) get a computer aided/holographic nursery for their children. The mother becomes increasingly agitated as the children withdraw more. The children are barely even featured in the story, and the ending concludes with lions possibly eating the parents.   Publishing date: Years ago.   Tech level: Seemingly functional AI, not much more. I did not know that you could find an answer for that. However, I can only use Excel so far to do it.  How to calculate 3.6! by hand? Geometrical representation of Contravariant and covariant vectors After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, which is contravariant to its basis. However, using the metric tensor, we can map this vector to a one form in the dual space, which acts as a vector space for this example. This one form, or covactor, varies similarly to the basis in our original vector space. However, the covectors have a different basis, which is the dual basis, and are expanded in terms of those. Here's where my confusion arises : Many diagrams over the internet give a geometric picture of the scenario, by claiming that contravariant components of an 'arrow' are found by drawing parallel lines along the axes, and checking where they intersect. Co-variant components could be found by sketching the perpendicular lines on these axes. So, these materials are treating covariant and contravariant as different representations of the same arrow, while I'm inclined to believe they are completely different. However, if we let go of rigour, in exchange for a 'more' geometric understanding, I think we are allowed to do this. The diagrams are usually of the form :  However, even if we assume that contravariant and covariant are different representations of the same object, this particular diagram still seems wrong to me. Particularly the locations of $x_1,x_2$. In this diagram, they are located on the span of the original basis. Shouldn't covector components be located on the span of the dual basis ? This diagram suggests to me, that co-variant components have the same basis, as the contravariant ones. Shouldn't the diagram look more like this ?  This second diagram seems to fit the concept better according to me. This is because the dual vector must be 'contravariant' with respect to the 'dual basis'. This means, their components must be found out by sketching parallel lines along the span of the dual basis. These lines intersect the original axis at a right angle, which is to be expected, as dual bases are orthogonal to the original bases. Moreover, this second picture can also show in a much better way, how scaling up of the original basis, scales down the 'contravariant components' and the 'dual basis', which in turn scales up the 'covariant' components.This shows that the components of the dual vector are covariant to the original basis. This is something that is not readily visible in the first diagram. So, am I correct in assuming this second 'geometrical' representation is correct ? I know this doesn't make much sense, because as the mathematicians said, that vectors are completely different from one forms, and one forms should be represented using the number of hyperplanes intercepted by our arrow. However, I've seen most course material refer to it in this way, and frankly it is easier to visualize. However, most of this material used the first picture. Can anyone point out my mistakes, if any, and tell me if the first picture is correct, or the second one ? Or in this case 'less-flawed'. Solving $\cos x=x$ I would like to know how can we solve the equation $\cos x = x$, without graphing. I know that  there would only be one solution, that is obvious, that too in between $0$  and $\frac{\pi}{2}$. Is there any real expression in finite terms [perhaps we call that closed form, I am not sure] that could give $x$ or $\cos x$. Although I have not studied Taylor series, I know that it only gives an infinite series, which is not what I want. I suspect that it could not be done, but can anyone explain me why?  Just for completeness, Wolfram Alpha, gives the approximate answer $x = 0.7390851332151606416553120876738734040134$, but fails to give a exact solution. Let us view our deleted comments If moderators can view , can we see comments we've deleted? We can see our deleted answers, but why not our deleted comments? I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.  The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem.   The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:   "The proof must be wrong."  "You cannot achieve such a result."  "You do not understand well the notion of ..."   My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes,  what should you do? "Its" and "it's" - how to remember? What is a good mnemonic rule that an English learner can use to remember the difference between its (possessive adjective: a team has started its lunch) and it's (verb: it's raining)? Why does the classical path give the dominant contribution in the path integral? C++ handling very large integers I am using the RSA Algorithm for encryption/decryption, and in order to decrypt the files you have to deal with some pretty big values. More specifically, things like   P = C^d % n   = 62^65 % 133   Now that is really the only calculations that ill be doing. I have tried using Matt McCutchen's BigInteger Library, but I am getting a lot of compiler errors during linking, such as:  encryption.o(.text+0x187):encryption.cpp: undefined reference to `BigInteger::BigInteger(int)'  encryption.o(.text+0x302):encryption.cpp: undefined reference to `operator&lt;&lt;(std::ostream&amp;, BigInteger const&amp;)'  encryption.o(.text$_ZNK10BigIntegermlERKS_[BigInteger::operator*(BigInteger const&amp;) const]+0x63):encryption.cpp: undefined reference to `BigInteger::multiply(BigInteger const&amp;, BigInteger const&amp;)'   So I was wondering what would be the best way to go about handling the really big integers that come out of the RSA Algorithm.  I heard that a possibility would be to declare your variables as a double long, so...  long long decryptedCharacter;   but I'm not sure exactly how big of an integer that can store.    Well for example, I try to compile and run the following program using dev C++:  #include iostream  #include "bigint\BigIntegerLibrary.hh"  using namespace std;  int main() {     BigInteger a = 65536;     cout &lt;&lt; (a * a * a * a * a * a * a * a);     return 0; }   then I get those errors.  Derek, I thought that by including the BigIntegerLibrary.hh file, that the compiler would go through and compile all the necessary files that it will use.  How should I try and compile the program above in order to resolve the linking errors? How can I vertically align elements in a div? How to animate only the camera? So  i have a scene with a few objects, and a water simulation in the scene. However I want to animate the camera turning around in the scene, while the water simulation is frozen in time. So something like a time freeze effect. But when ever I try to render an animation it renders the water simulation too. I just need one frame of the water simulation. Is there a way to do this? Hydrogen radial wave function infinity at $r=0$ When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{\ell,m}(\phi,\theta).$$ I understand that the radial part usually has a singularity for the $1s$ state at $r=0$ and this is why you remove it by writing: $$R(r) = \frac{u(r)}{r}$$ But what is the physical meaning of $$R(r=0) = \infty~?$$ Wouldn't this mean that the electron cloud is only at the centre of the atomic nucleus? Why was Sirius ever convicted? "Reality" of length contraction in SR I was in argument with someone who claims that length contraction is not &quot;real&quot; but only &quot;apparent&quot;, that the measurement of a solid rod in its rest reference frame is the &quot;real length&quot; of the rod and all other measurements are somehow just &quot;artificial&quot; and &quot;apparent&quot;. Seemed to me like a bad conversation about poorly defined words and not really relevant to physics, but then some quotes were offered:  ...so that the back appears closer to the front. But of course nothing has happened to the rod itself.  (Rindler)  The effects are apparent (that is, caused by the motion) in the same sense that proper quantities have not changed.  (Resnick &amp; Halliday) At the same time, the  insists that length contraction and time dilation are &quot;real&quot;, with observable consequences:  Time and space dilation are often referred to as ‘perspective effects’ in discussions of STR. Objects and processes are said to ‘look’ shorter or longer when viewed in one inertial frame rather than in another. It is common to regard this effect as a purely ‘conventional’ feature, which merely reflects a conventional choice of reference frame. But this is rather misleading, because time and space dilation are very real physical effects, and they lead to completely different types of physical predictions than classical physics. [...] However, this does not mean that time and space dilation are not real effects. They are displayed in other situations where there is no ambiguity. One example is the twins' paradox, where proper time slows down in an absolute way for a moving twin. And there are equally real physical effects resulting from space dilation. It is just that these effects cannot be used to determine an absolute frame of rest.  I went through a lot of Einstein, Minkowski and Lorentz original material, and didn't find anything about what is &quot;real&quot; and what is not. Finally, I know about muons, where the effects of SR seem to be very real (from , but I had seen it in a physics class before):  When a cosmic ray proton impacts atomic nuclei in the upper atmosphere, pions are created. These decay within a relatively short distance (meters) into muons (their preferred decay product), and muon neutrinos. The muons from these high energy cosmic rays generally continue in about the same direction as the original proton, at a velocity near the speed of light. Although their lifetime without relativistic effects would allow a half-survival distance of only about 456 m (2,197 µs×ln(2) × 0,9997×c) at most (as seen from Earth) the time dilation effect of special relativity (from the viewpoint of the Earth) allows cosmic ray secondary muons to survive the flight to the Earth's surface, since in the Earth frame, the muons have a longer half life due to their velocity. From the viewpoint (inertial frame) of the muon, on the other hand, it is the length contraction effect of special relativity which allows this penetration, since in the muon frame, its lifetime is unaffected, but the length contraction causes distances through the atmosphere and Earth to be far shorter than these distances in the Earth rest-frame. Both effects are equally valid ways of explaining the fast muon's unusual survival over distances.  So which is which? Why do Rindler, Resnick &amp; Halliday use the word &quot;apparent&quot;? "any other verbs then being located in VP" -- is this acceptable? if the sentence you are analysing contains a modal verb, then it is positioned in ‘I’,any other verbs then being located in VP.      (English Syntax and Argumentation - Bas Aarts)   At first, I think the The bold clause should be "...is located..." but then I realize it might be in the form of a non-finite clause (ing clause with a subject) but I'm not sure. If it isn't, how you describe this usage? Is this common in English?