Do the virtual particles have any mathematical meaning in vacuum in absense of real quanta? I have edited this question with its title. Virtual particles, as I understand, arise as intermediaries in the calculations describing interactions such as scattering between real particles and in drawing certain Feynman diagrams.   My question is not whether virtual particles have real physical existence by direct observation. My question is, whether the mathematical language of "virtual particle-antiparticle popping in and out of existence", is even mathematically meaningful whenever you take everything (all real quanta) out of space. By empty space, I understand, the vacuum of a free or an interacting quantum field theory. May be  roughly explains my question. Can the vacuum loops be given such a meaning?  Edit-The maximum voted answer in  talks about virtual particles in real scattering experiments. But it does not address whether this virtual particles have any mathematical meaning, when there are no real quanta or particles present in the picture.
Do virtual particles actually physically exist? I have heard  pop in and out of existence all the time, most notable being the pairs that pop out beside black holes and while one gets pulled away. But wouldn't this actually violate the conservation of energy?
Could the Olympians use modern celestial weapons? Instead of using short range celestial bronze (CB) weapons like swords and arrows why not make advanced weapons like machine guns or grenades with CB shrapnel in it? Can't read filenames in save dialog box. Names are too faint I want to click on file to inherit the name for the file being saved. I can't read file names in the dialog box because the color is too faint. This is the same question as   That question does not have an answer but I don't have enough points to interact there so I have to post a new question.    Edited to add screen shots I tried to calibrate the monitor which no effect and did not change the readability of "unavailable" file names. I have added screenshots of previous operating systems to compare: Please tell me how Mojave is not to blame here: CakePHP View including other views I have a CakePHP application that in some moment will show a view with product media (pictures or videos) I want to know if, there is someway to include another view that threats the video or threats the pictures, depending on a flag. I want to use those "small views" to several other purposes, so It should be "like" a cake component, for reutilization.  What you guys suggest to use to be in Cake conventions (and not using a raw include('') command) Not able to understand the meaning of question? car moving with a speed of 40 km / h can be stopped by applying brakes after at least 2 m. If the same car is moving with a speed of 80 km / h, what is the minimum stopping distance  Does this mean that after 2 m, You will apply brakes and then it stops. What is the retardation value  with which you are brakes.I am not understand the meaning of the values. Finding $\lim_{n\to\infty}\int_0^{\frac{1}{2^n}}(1+\cos(x))^n\,dx$ During our studies on Riemann Integrals we were given the following question in my Calculus II course: $$ \forall n\in\mathbb{N}: \text{let } f_n(x)=(1+\cos(x))^n, f_n:[0,1]\to \mathbb{R}$$ $$\text{ Find}: \lim_{n\to\infty}\int_0^{\frac{1}{2^n}}f_n(x)\, dx $$ I have tried finding the limit using the squeeze theorem. I noticed that the function is defined on $[0,1]$, where $\cos(x)$ is positive and decreasing. I used the following boundaries: $$ 1\leq f_n\leq 2^n \implies \frac{1}{2^n} = \int_0^{\frac{1}{2^n}}1 \leq \int_0^{\frac{1}{2^n}}f_n(x) \leq \int_0^{\frac{1}{2^n}}2^n = 1  $$ but $ \displaystyle{\lim_{n\to\infty}}\frac{1}{2^n} = 0 $ which means I'm unable to apply the squeeze theroem. I've tried using tighter lower boundaries such as $1+\cos(1)$ But it didn't seem to work. I did some trial and error by calculaing the integrals manually for different values of $n$ and it does seem like $\displaystyle{\lim_{n\to\infty}}\int_0^{\frac{1}{2^n}}f_n(x) dx = 1 $ This means that somehow I should find a lower bundary which tends to 1 when $n\to\infty$. Any ideas on how to go forward from here are welcome. If $f^2$ and $f^3$ are smooth, does it follow that $f$ is smooth? Let $f: \mathbb{R} \to \mathbb{R}$ be given. Assume that the square and cube of $f$ are smooth. Is $f$ smooth? That is if $f \cdot f \in C^{\infty}$ and $f \cdot f \cdot f \in C^{\infty}$, does it follow that: $f \in C^{\infty}$   I got this from another question on SE.   So, $f(x)^2$ is infinitely differentiable, and so is: $f(x)^3$   Also, realize:  $$f(x) = \frac{f(x)^3}{f(x)^2}$$  But what can I do? I am not able to play "Half-Life-2" on Steam + Ubuntu 14.04 . Error shown in message box on screen:      Could not find required OpenGL entry pint 'glGetError'! Either your video card is unsupported, or your OpenGL driver needs to be updated.   I have a Dell Inspiron Laptop.  My OS is Ubuntu 14.04 and my graphics card is AMD Radeon HD 8600M.  I have installed AMD properietry Driver.    The partial output of "glxinfo" is as follows:      name of display: :0     libGL: screen 0 does not appear to be DRI3 capable     libGL: pci id for fd 4: 8086:0a16, driver i965     libGL: OpenDriver: trying /usr/lib/i386-linux-gnu/dri/tls/i965_dri.so     libGL: OpenDriver: trying /usr/lib/i386-linux-gnu/dri/i965_dri.so     libGL: Can't open configuration file /home/arushi/.drirc: No such file or directory.     libGL error: failed to load driver: i965     libGL: OpenDriver: trying /usr/lib/i386-linux-gnu/dri/tls/swrast_dri.so     libGL: OpenDriver: trying /usr/lib/i386-linux-gnu/dri/swrast_dri.so     libGL: driver does not expose __driDriverGetExtensions_swrast(): /usr/lib/i386- linux-gnu/dri/swrast_dri.so: undefined symbol: __driDriverGetExtensions_swrast     libGL: Can't open configuration file /home/arushi/.drirc: No such file or directory.     libGL: Can't open configuration file /home/arushi/.drirc: No such file or directory.     libGL error: failed to load driver: swrast     display: :0  screen: 0     direct rendering: No (If you want to find out why, try setting LIBGL_DEBUG=verbose) Uxie and Azelf Are Not Appearing in Nameless Cavern How do I uninstall xcode 5 command line tools? 'find' utility does not output all files when using wildcards Running:  $find . -name *.exe   gives:  ./MakeItSo_1.2.2/MakeItSo_1.2.2/MakeItSo.exe ./MakeItSo.exe   but those are not the only exe files in the directory. For example, running  $find . -name ATLTester.exe   gives:  ./Debug/ATLDmoVexaTester.exe lognormal distribution, standard-deviation and (physical) units I have a distribution of microparticles that follows a lognormal distribution. The cumulative distribution function thus is given by:  $$ F_X(x;\mu,\sigma) = \frac12 \operatorname{erfc}\!\left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) $$ $$ \mu = ln(M) + \sigma^2 $$  Now, the plot of the distribution function should be exactly the same no matter if the particle diameter $x$ is given in micrometers or meters (as long as I adapt the x-axis accordingly of course). However, this only works if I only convert $x$ and $M$, while not touching the numerical value of $\sigma$, and I don't understand why. $F_X$ has to be unitless, so $x$, $\mu$ and $\sigma$ should all have the same unit, right? Our high-school teacher told us that the Earth pulls us with some force $F$ and we pull the Earth with the same force $F$. Within Newtonian physics this is true because of , but let’s consider . My mass is small; so I don’t warp space-time much. But Earth’s large mass warps space-time to a far greater extent.  So do I pull Earth with the same force it pulls me? If yes, how? Youtube says "This video is currently unavailable" no space left in '/' but a lot space left in '/home' i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' Whenever I try to install anything with the Ubuntu Software Center, the following error appears.     Failed to download repository information      Check your Internet connection.      E:Unable to parse package file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_precise_multiverse_i18n_Index (1) I have been invited to write a book chapter in an edited volume, relating to my field of work. I have a paper ready that I have not published and was planning to submit it soon. So, I am wondering is it worth to use that article for the book chapter, as both are on the same topic, or is it in a sense a waste of a journal paper, as book chapters might not be seen to have the same academic value? What should I do? Should I, instead, send the ready paper to a journal for a review and write a book chapter, based on a couple papers I have published, for instance?  Generally speaking, where does a book chapter stand? Is it something between a conference paper and a full-fledged journal article? Or, does does it have the same value as either of the above-mentioned publication types, in your opinion? Cannot get specific WMS to display in Leaflet or OpenLayers 3 Considering this code, can I be absolutely sure that the finally block always executes, no matter what something() is?  try {       something();       return success;   }   catch (Exception e) {        return failure;   }   finally {       System.out.println("I don't know if this will get printed out"); } Geometric interpretation of Electromagnetism I have forgotten my ID proof and I'm doing an Indian domestic flight. Can I still fly? I am travelling from Delhi to Indore, a domestic flight. I forgot to take ID proof. Can I still fly?