In regular math, you cannot get the square root of a negative number. Likewise, you cannot divide by zero. Both dividing by zero and taking the square of a negative have no place in real life.  However, we have systems in place to deal with the square root of a negative number (hence, imaginary numbers). Why does an option for divide by zero not exist?
We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers).  Why this trick isn't performed with division by zero?
$\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$ How to add “jQuery” onclick functionality for a column each row in jQgrid to display different popups for different rows C++ virtual function from constructor Why the following example prints "0" and what must change for it to print "1" as I expected ?  #include &lt;iostream&gt; struct base {    virtual const int value() const {       return 0;    }    base() {       std::cout &lt;&lt; value() &lt;&lt; std::endl;    }    virtual ~base() {} };  struct derived : public base {    virtual const int value() const {       return 1;    } };  int main(void) {    derived example; } How to show the parent and child taxonomy term? how to read huge csv file into R by row condition? The  is a proposed propulsion mechanism for spacecraft in which some form of microwave device provides the propulsion. The consensus from the physics community, including many voices on this site, is that the proposed device would violate conservation of momentum.   Several experiments have been conducted, by three main groups: (i) , from Satellite Propulsion Research Ltd, (ii) a research group at the Northwestern Polytechnical University in Xi'an, China, and (iii) Eagleworks Laboratories at NASA's Lyndon B. Johnson Space Center in Houston, Texas. However, few of these experiments have been performed in vacuum (and none of the vacuum tests has been published in a peer-reviewed journal), which means the effect of air currents around the device cannot be ruled out.  This device has been the subject of many questions on this site, including          and their many closed duplicates.  Many of these questions ask what the proposed explanations are, but then veer off into territory of the form "assuming it is true, how does it work?", so they have been (rightfully) closed as counterfactual physics. I feel this sort of bundling does a lot of harm to the discussion, and there is some scope to ask what exactly is being proposed as an explanation. So:     What do the proponents of EmDrive claim is the theoretical justification of the device's function?   Please limit answers exclusively to references, and indicate whether they are peer-reviewed or not. Please limit answers to serious attempts at explanations, as opposed to vague claims about the 'quantum vacuum' - or indicate if an experimental paper does not have any such theoretical explanation or reference.  To be clear, this is not the place to discuss whether the explanations are correct or not. If you have specific questions about any of these sources, ask it separately. (However, if your question starts with "assuming that this works", be prepared for it to be closed.) If you feel there are not enough resources on this thread, you can  but please do not post another question asking people to look for more references. To be clear, I have access to $H(S)$ and as many $H(S||K)$ as I want. Every string $K$ is known. Can I use the extra information to my advantage and try to deduce $S$, without requiring bruteforce? Or do you still have to bruteforce but try less combinations since you have more information? "Above"/"below" before/after a noun I have seen sentences similar to the following:  (1) See the reference above. (2) See the reference below.  And,  (3) See the above reference.  But not,  (4) See the below reference.  Are all these forms acceptable? Which is/are preferred in formal writing? Side-by-side source and output when documenting a style file I checked the other question similar to this issue yet I could not find a proper solution for the issue I am facing for including an .svg file which is converted using inkscape with pdf+latex option set to ON, and I have also included both the .pdf and .pdf_tex file in the path. Yet I get the above mentioned error.   My code:  \begin{figure}[h!]  \centering \def\svgwidth{\columnwidth} \input{Implementation/Figs/most_changed_from_default_pos_0.pdf_tex} \end{figure}   Full error description:  !pdfTeX error: pdflatex (file /compile/Implementation/Figs/most_changed_from_default_pos_0.pdf): PDF inclusion: required page does not exist &lt;5&gt;   Any help to solve this issue would be really appreciated. I also downloaded the pdf file to check whether it has 5 pages. I could verify it, there were 5 pages in the pdf file.   The file which I want to convert is   I have added the google drive link to the file which I have used for analysis herewith I see more and more servers that have no domain defined in the /etc/resolv.conf file.  Only a search entry. In what cases is this setup used ? Are there any cases where it must not be used ?  nameserver xxx.xxx.xxx.xxx nameserver yyy.yyy.yyy.yyy nameserver zzz.zzz.zzz.zzz  search domain1 domain2 domain3 ... Bind variable with dotted name not working in dynamic SOQL As mentioned in , it's possible to use bind variables in dynamic SOQL queries.  This seems like a great way to avoid SOQL-injection, and to build a query string once but use it multiple times with different values in the variables.  I'm working on a method that returns a list of objects.  It accepts a class with multiple properties as its only parameter.  The properties of this class are used to filter the results to just the required records.  It will eventually be driven by a Visualforce component, which will be configured by querystring parameters in the URL.  I just ran into an issue where the query fails when I use dotted bind variables.  If I copy the constructed dynamic SOQL into a static query, it works fine.  This may be a bug, because in doing tests I found that :foo.bar would actually try to use local variable bar.  Here's one of my queries, which uses related custom objects.  I've stripped it down to just the part that's failing (wrapped for legibility):  result = Database.query('SELECT Name, Another_Object__r.Name,      Another_Object__r.Custom_Field__c      FROM Custom_Object__c       WHERE Another_Object__r.Custom_Field__c = :filter.deviceType');   The error reported is System.QueryException: Variable does not exist: filter.deviceType.  As mentioned above, the query works fine as static SOQL, just not dynamic:  result = [SELECT Name, Another_Object__r.Name, Another_Object__r.Custom_Field__c FROM Custom_Object__c WHERE Another_Object__r.Custom_Field__c = :filter.deviceType];   It also works in this case:  String deviceType = filter.deviceType; Database.query('SELECT Name, Another_Object__r.Name, Another_Object__r.Custom_Field__c FROM Custom_Object__c WHERE Another_Object__r.Custom_Field__c = :deviceType');   I haven't read about this being a known limitation.  I don't know if I should be using some other syntax/notation in the query -- the dotted notation appears correct.  Time to report a bug?  Thoughts on workarounds other than creating local variables to stand in for each member of the class? Is RSA key size the size of private key exponent? Solvable Lie algebra with codimension 1 ideal There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"? What is a magic number, and why is it bad? Cannot Buy Land in Falkreath I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks How does a  work, and what differentiates it from other linear classifiers, such as the , , or ? *  (* I'm thinking in terms of the underlying motivations for the algorithm, optimisation strategies, generalisation capabilities, and run-time complexity) The  specifically says not to ask  for career advice but people do it anyway.  Should I consider the Help Center outdated or only loosely enforced?    And what about a career advice site on the network?  There are 70+ sites, and most of them are for relatively narrow interests (which is not a criticism, I enjoy looking through the many sections).  A career section seems like it would be very popular.  Also, a careers section would help keep career questions off the other sites. How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?  This should be easy, but I'm not seeing how.  A hint would be appreciated.