Why are so many energies in our universe mathematically represented by the equation $\frac{1}{2}ab^2$.  For example:    Kinetic energy $$\frac{1}{2}mv^2$$ Energy stored in a capacitor $$\frac{1}{2}CV^2$$ Energy stored in an inductor $$\frac{1}{2}LI^2$$   These are just some I can remember off the top of my head ,I remember that there are more like this. Is this just a coincidence or is there a reason behind this ?  I tried to google it but couldn't find any relevant results.
In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.  The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.  The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.  The formula for energy stored in a capacitor is $\frac{1}{2}C \Delta V^2$.  The formula for energy delivered to an inductor is $\frac{1}{2}LI^2$.  Finally, everyone is aware of Einstein's famous formula $e=mc^2$.  I realize there are other energy formulas (gravitational potential energy, for example) that do not take this form, but is there some underlying reason why the formulas above take a similar form? Is it a coincidence? Or is there a motivation for physicists and textbook authors to present these formulas the way they do?
https:// search results appearing on Google for purely http:// site I am trying to create a vase that looks similar to this:    If there is another post out there I will happily remove this one or redirect it there.  Specifically, I am trying to create the creased effect in the geometry that looks like it is twisting around the vase.  I have tried using Checker Deselect and extrude->scaling the verts but that really screws up my mesh.  Link to my model if it helps: How to input a specific Pantone color number in Illustrator? I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The  contains a delta function in the formula, but  gives only the formula without the delta function,  $$\bar{E} = -\nabla \phi = \frac{-p}{4\pi \epsilon_{0} r^{3}} ( \hat{e}_{z}-3\hat{u}_{r}(\hat{u}_{r} \cdot \hat{u}_{z}) ) $$  Is it intentionally left out?  My notes mention that the dipole is with a circular thing $P=Q d$ where $d$ is distance and $Q$ is charge. Possible Duplicate:          Here is a test case:   void foo(int i, int j) {    printf("%d %d", i, j); } ... test = 0; foo(test++, test);   I would expect to get a "0 1" output, but I get "0 0"  What gives?? What are examples of two non-similar invertible matrices with same minimal and characteristic polynomial and same dimension of each eigenspace? Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ Show that every prime $p&gt;3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$ I'm looking to do a third-party assessment of the false-positive rate of a video classification algorithm. Since I have a lot of video I'm trying to do a power analysis to figure out exactly how much video I need to look through so that it is representative of all the video data at a given confidence interval.  The algorithm flags video sequences that have at least one cat in it, and I'm looking to evaluate the frequency of false positives on a new unlabeled test set. So I have tagged all the video that my algorithm has identified a cat in and now want to sample the tagged video sections and look through them manually to validate my model since looking through all of it would take too long! Note, I'm not looking to refine the model at this point, just assess it.  My null hypothesis is that the FP rate of the sample of video I watch is equal to the FP rate of all the video.   I think I can use this formula to determine the number of video sequences to view:    Here is my question: am I thinking through this formulation correctly? Since my model has a CV false positive rate of ~0.96, I figure I can use that as a reference. Can I use that for the null hypothesis proportion, p0? Or will that be p, the true proportion?    I've been using this online calculator:   I ask because when setting the parameters I have, I'm getting very small sample sizes, like less than 10 sequences to view. That can't be right. Square bracket notation for dimensions and units: usage and conventions One of the most useful tools in  is the use of square brackets around some physical quantity $q$ to denote its dimension as  $$[q].$$  However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them? I am having the following equation:  \begin{equation}   Q(\lambda,\hat{\lambda}) = -\frac{1}{2} P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) \left( n \log(2 \pi ) + \log \left| C_m^{(s)} \right| + \left( \mathbf{o}_t - \hat{\mu}_m^{(s)} \right) ^T C_m^{(s)-1} \left(\mathbf{o}_t - \hat{\mu}_m^{(s)}\right) \right) \end{equation}   which does not very well fit on one line. How can I split this over two lines? What I have in mind is that I specify the splitting place, and that the first line is left aligned and the second line right aligned to make clear that it is still the same equation.  The linebreak \\ does not work. What is the cause of gravitional force of attraction? Ok,this is a silly question. But for quite long I have been thinking about this. What is gravitation all about?? My book directly writes since force of gravitation is directly proportional to the masses and inversly proportional to the square of the distance ie. $$F = G\frac{M_1 . M_2}{r^2}$$ . So is this gravitation all about? What is the cause of it? The book then says     We are not in the scope in classical approach so as to discuss the cause of this. However,as a matter of fact that if there are more than one mass,then there is gravitation as it is their fundamental property.( Cause of gravitation can be attributed to exchange of non-classical particles between the masses) .    Then it ended the talkings. What are those non-classical particles? Why and how are they exchanged? So far I have studied physics , I never found such weird phenomenon like this. I want to know what is the cause of gravitation and what is the cause of exchange of those so-called non-classical particles? And how do they do it? Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. The statement:  For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n - 1$.  Using induction over $n$, the base step comes easy since $m^n - 1$ is $0$ when $n = 0$.  My induction hypothesis is letting $k \geq 0$ and assuming that $m - 1$ divides $m^k - 1$. In order to show that $m - 1$ divides $m^{k+1} - 1$, I obviously need to use the induction hypothesis. However, no matter where I try to use the fact that $m^k - 1 = (m-1)a$ for some $a$ in the integers, the expression $m^{k+1} - 1$ always becomes more difficult to get to $m^{k+1} - 1$ being equal to $(m-1)b$ for some $b$ in the integers.   In other words, I can't figure out any actually helpful way to apply the induction hypothesis with the goal of proving the next step.  Any help would be appreciated! Since i upgraded to Ubuntu 15.04, if i click on the shutdown button it only logs out and i have to click again on the shutdown button (from the ubuntu login screen) to power off the pc. How can i fix this? How can I generate irregular grid containing minimum n points? Given a large (~ 1 million) sample of unevenly distributed points - is it possible to generate irregular grid (in size, but could also be irregular in shape if that is possible?) that will contain specified minimum amount of n points?   It is of less importance for me if genereted 'cells' of such grid contain exactly n number of points or at least n points.  I'm aware of solutions like  in ArcGIS or  in QGIS/mmgis however they will all create regular grids which will result in an output with empty cells (smaller problem - I could simply discard them) or cells with count of points less than n (bigger problem since I'd need a solution to aggregate those cells, probably using some tools from ?).  I've been googling around to no avail and am open to both commercial (ArcGIS &amp; extensions) or free (Python, PostGIS, R) solutions. Use Results of Find Command to Call vi Is there a way that I can pipe the output of this command (assuming there is only one file found) to vi.  I want to find this command, and then call it up in vi.  Thanks.  find . -name 'id.properties' How Does Kylo Ren not stop Chewbacca's blaster? How does Kylo Ren stop the blaster shot in the beginning when being shot at by Poe, only to be unaware of the coming blaster shot from Chewbacca, which seems much more predictable.   I am sure he knew Chewbacca, Rey and Finn where there as Chewbacca turns with the opening of the doors when Rey and Finn come in.  EDIT in regards to Richards Answer: I just want to expand a little more on question. What I don't understand is that Kylo Ren uses the force, so it is just not a matter him not hearing. When Poe shoots him from behind, he makes no sound, but yet he turns and stops the blaster and Poe. That has to be due to the power he commands in the Force.  And this is from Obi-Wan Kenobi:     "Well, the Force is what gives a Jedi his power. It's an energy field   created by all living things. It surrounds us and penetrates us; it   binds the galaxy together."―Obi-Wan Kenobi —   So it seems to me that it would be very hard for him even in a weakened state to not even notice that he was being shot at by virtue of the force being a part of Sith and Jedi's alike. Even if he was too weak to actually stop the blaster, the Force should have been able to warn him. TikZ doesn't fill when node coordinates are used Can someone please explain me why the following code doesn't result in a fill?  And why, sometimes, I get such "very thin lines" artifact (that's visibile in my picture)?  \begin{tikzpicture} \node (A) at (0,0) {A}; \node (B) at (1,0) {B}; \node (C) at (1,1) {C}; \fill[red] (A) -- (B) -- (C) --cycle; \end{tikzpicture}   Here's what I get: List comprehension with expensive computation Upgrading straight handlebars to drop handlebars I have a two-and-a-half year old  (with a road-bike-like frame and thin tires; it's somewhere in the loosely-defined "hybrid" range) that I use to commute to work, twenty-five miles round-trip, once or twice a week. The bike has straight handlebars, and having gotten used to the idea of commuting via bike, I'm interested in upgrading to drop handlebars. It looks like I'd need new break/shift levers to go with the handlebars; the whole upgrade looks like it'd run a couple hundred dollars. This is a sizable fraction of the original cost of the bike, which leads me to wonder if I'm better off saving my money for a new bike that's a little better adapted to my use.  Is upgrading my bike's handlebars a reasonable investment? Given the size of the Empire, it seems highly probable that names would be reused. Although I'm not aware of any in canon, there must be other Skywalkers in the galaxy, leading me to my question.  Are there other Skywalkers in the galaxy, and are they mentioned in canon? Does Luke have any third cousins twice removed (etc.)?  Bonus points if you can attest to force sensitivity among the surname.