Which fundamental force is responsible for the Casimir effect?
what is the relationship between the dynamical casimir effect and virtual particles?
I have heard of some methods, but none of them have stuck. Personally I try to avoid complex types in C and try to break them into component typedef.  I'm now faced with maintaining some legacy code from a so called 'three star programmer', and I'm having a hard time reading some of the ***code[][].  How do you read complex C declarations? Fundamental group of a torus with points removed Methods to work around the problem of missing data in machine learning Problem with option border of class standalone I tested a MWE given in question . Following the documentation (page 9) I used the class option border={left right bottom top} with left=right=top=0pt and bottom=20pt (20pt only to show the effect here).  With the version 2012/09/15 v1.1b of standalone and the following MWE I get no 20 pt margin at the bottom of my picture.  What is the problem?  MWE:  \documentclass[% % border=1pt   border={0pt 0pt 20pt 0pt} %left right bottom top ]{standalone}% http://ctan.org/pkg/standalone \begin{document} $2^5 = x_5 \times y^8$ \end{document} Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $ Where $ f: V \rightarrow V  $ is a linear operator on an inner product space $V$, I'm trying to show that $$ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $$  i.e. the Kernel of the adjoint is equal to the orthogonal complement of the image of f.  I taking the inner product of $f(v)$, where $v \in V$  and an arbitary element of the kernel, $k \in \mathrm{ker}(f^*) $  $$\langle f(v), k\rangle = \langle v, f^*(k)\rangle = \langle v, 0\rangle = 0$$  And then taking the inner product of $f(v)$ with $w$, where $w \in  \mathrm{im}(f)^{\perp}$  By definition  of the orthogonal complement to the image of f,   $$ \langle f(v), w\rangle = 0 $$  My question is, now that we have   $$ \langle f(v), k\rangle = \langle f(v), w\rangle = 0 $$  What can we conclude? I would like to conclude that because $w$ and $k$ are arbitary elements of their respective subspaces, that these must be equal, i.e. $ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $  However, I am not sure this is mathematically sound, is that conclusion valid or are there any technical barriers or additional statements that would have to be made to make that watertight? Is there any difference in equals and == for String variables? if ('text'.equals(str))   or   if ('text' == str)   Is there any difference? If yes, then what is it?   Added, after answered.  String q = 'teXt'; system.debug('teXt'.equals(q)); // true system.debug('teXt' == q);      // true system.debug('text'.equals(q)); // false system.debug('text' == q);      // true Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$? Error when updating texlive 2015 When I want to update packages of texlive 2015 I get this error? what is the problem?  tlmgr update --list  The locale codeset (cp720) isn't one that perl can decode, stopped at   C:\texlive \2015\tlpkg\tlperl\lib/Encode/Locale.pm line 123. Compilation  failed in require at C:\texlive\2015\tlpkg\tlperl\lib/LWP/UserAgent. pm  line 1001. I've installed phpmyadmin sudo apt-get install phpmyadmin. Of course, I have also mysql and a web server running properly.  Now how can I reach it? Would Beetee's electrocution plan work and kill the tributes in the lake? In , the second episode of the Hunger Games trilogy, one of the characters (Beetee) proposes a plan to kill some of the remaining tributes. He proposes wrapping a special wire that he has designed to withstand the vast energy of lightning around a tall tree and running it down to a salt water lake. Then when lightening strikes the tree (which it does predictably every twelve hours) anyone in the lake will be killed. (N.B. This is not Beetee's real plan, but it is the one he wants his allies to believe.)  I don't think his plan would work at all. Assuming that his wire can carry the enormous current delivered by a lightening strike (this is fiction after all) surely the current would short to ground through the salt water rather than taking a path through anyone standing or swimming in the water. The analogy I have in mind is that of birds standing untroubled on live electricity wires. They are unharmed because they do not provide a path to earth. But another analogy might be dropping a mains electric heater into a bath when you are in it. I think that kills you (though I'm not sure why.) Sigma algebra and algebra difference How can I find the product key of Office 2011 after restoring from backup? I had to restore my iMac from a Time Machine backup.  Everything seemed fine until I tried to start Office 2011 and it is asking me to re-enter my product key. I am trying to locate my key (I believe I downloaded this version originally and it should be in my email but I can't find it)  I tried a program that is supposed to "find the keys for products installed on your computer" (Mac Product Key Finder) and it found my very old Office 2008 key, but not 2011.  I am surprised that if I had a totally working install before recovering from back-up that this wasn't restored as well - do I have any options to find it hiding somewhere on my drive?  Note: The recovery was after a hard drive was replaced by the Apple store - if that makes a difference, so this is a new internal drive. How can I set Excel to always import all columns of CSV files as Text? Though I try to avoid it, I occasionally have to open a CSV file in Excel. When I do, it formats columns containing numbers, which makes them useless for my purposes. As far as I can tell, the only way to prevent this from happening on import is to rename the file so the extension isn't .csv and use the import wizard to specify the format of each column individually. For files with 50-60 columns, this is impractical.  Since every answer for this oft-asked question on the internet suggests either some means of converting the formatted numbers back once the file is open (which won't work for me - I want to solve the general problem, not a few specific cases) or manually selecting the format type of each column (which I don't want to do), I'm looking for a way to set a global preference or style such that all columns of all CSV files opened are always formatted as text.  I know about "armoring" the numbers with quotes, too, but the files I get don't come like that and I was hoping to avoid having to pre-process the files so Excel doesn't screw them up.  Is there a way to do specifically this: Always format all columns in opened CSV files as text, without manually selecting each column every time during import?  I'm using Excel 2003, but I'll take answers for 2007 if that's what you know. Which "mean" to use and when? So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).  However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).  This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!  I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?" I've recently gotten into The Flash TV show, and so I've been reading about his powers. What I don't understand is why his typical rogues pose a threat - it seems like there should be very little limit to his powers.  He can go back in time, so if he loses a battle or the bad guys do something, he can go back and preempt it. In the new 52, he can all but predict the immediate future, so he can see bad things that are going to happen and stop them. He can phase matter through matter, or let matter pass through him. This means he is all but immune to weapons fire.  I haven't read enough of the old comics so I don't the know challenges he's had. For a guy who can't be attacked, can time-travel and can race around the globe several times in a minute, I can't imagine there is much that could pose a threat. I am flying from Orange County, California to San Francisco, California and then on to Seoul, South Korea. If I check a bag will I have to pick it up in San Francisco and re-check it and go through security or customs for the international flight?  Everything I am reading leads me to believe that it's only necessary to re-check baggage and go through security and/or customs when you are first arriving into a country. However, since my first leg is staying within the United States I assume the airline will transfer my bag to the next flight and I can just go straight to the gate for my international flight?  Both legs of the flight are through the same airline (United Airlines) and there are only the two legs: OC -> SF, then SF -> Seoul. How can I replace a string in a file(s)? Replacing strings in files based on certain search criteria is a very common task. How can I    replace string foo with bar in all files in the current directory? do the same recursively for sub directories? replace only if the file name matches another string? replace only if the string is found in a certain context? replace if the string is on a certain line number? replace multiple strings with the same replacement replace multiple strings with different replacements Please add TeX rendering on the Android app 'locate' command in terminal. Why is it so fast? Get Record Type Developer Name without SOQL Query