I'm quite confortable with definitions such as a derivative or an integral because i get to see why they are the way they are and that feels really natural, there is a fundamental question to be solved such as "What is the slope at a certain point of the curve" and so on...  I can't see why someone would come up with the idea of a determinant, don't tell me that gives some area , I don't see why giving an area or a volume solves the kind of problems determinants do, is there a nice line of reasoning for determinants? Thank you guys !
While I basically understand what a determinant is, I wonder how this idea was developed?  What was the principal idea behind its origination?  I would like to know this so that I can have a better conceptualization of the determinant.
How can an encryption scheme be indistinguishable with multiple messages but vulnerable to an adaptive chosen-plaintext attack? In order to be indistinguishable with respect to multiple messages, wouldn't the scheme need to be nondeterministic? How would this then possibly be vulnerable to an adaptive CPA? Galois group of irreducible cubic equation I'm writing code to do Xml serialization. With below function.  public static string SerializeToXml(object obj) {     XmlSerializer serializer = new XmlSerializer(obj.GetType());     using (StringWriter writer = new StringWriter())     {         serializer.Serialize(writer, obj);         return writer.ToString();     } }   If the argument is a instance of class without parameterless constructor, it will throw a exception.     Unhandled Exception:   System.InvalidOperationException:   CSharpConsole.Foo cannot be serialized   because it does not have a   parameterless constructor.    at   System.Xml.Serialization.TypeDesc.CheckSupported()   at   System.Xml.Serialization.TypeScope.GetTypeDesc(Type   type, MemberInfo sourc e, Boolean   directReference, Boolean throwOnError)   at   System.Xml.Serialization.ModelScope.GetTypeModel(Type   type, Boolean direct Reference)    at   System.Xml.Serialization.XmlReflectionImporter.ImportTypeMapping(Type   type , XmlRootAttribute root, String   defaultNamespace)    at   System.Xml.Serialization.XmlSerializer..ctor(Type   type, String defaultName space)    at   System.Xml.Serialization.XmlSerializer..ctor(Type   type)   Why must there be a parameterless constructor in order to allow xml serialization to succeed?  EDIT: thanks for cfeduke's answer. The parameterless constructor can be private or internal. $\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups I'm trying to solve this question of Hungerford's Algebra book:     $S_3$ is not the direct product of any family of its proper subgroups.   The same is true of $\mathbb Z_{p^n}$.   The first claim is easy: we note that every subgroup of $S_3$ is of order $2$ or $3$ by Lagrange, since its subgroups are of prime order, they are cyclic, then abelian, but $S_3$ is not abelian, contradiction because $S_3$ is not abelian while the direct product of abelian groups has to be abelian.  My problem is with the second claim, I need help. One of my coworkers has provided me with a Bash syntax that I am unfamiliar with.  My Google foo has failed me on figuring out what it does and why/when I should use it.    The command that he sent me was of this form:  someVariable=something command   Initially, I thought that this was equivalent to the following:   someVariable=something ; command   Or  someVariable=something  command   But this doesn't appear to the be case.  Examples:   [Jan-03 11:26][~]$ # Look at the environment variable BAZ. It is currently empty [Jan-03 11:26][~]$ echo $BAZ  [Jan-03 11:27][~]$ # Try running a command of the same format     [Jan-03 11:27][~]$ BAZ=jake echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:27][~]$ # Now, echo BAZ again. It is still empty:     [Jan-03 11:27][~]$ echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:28][~]$     [Jan-03 11:28][~]$ # If we add a semi-colon to the command, we get dramatically different results: [Jan-03 11:28][~]$ BAZ=jake ; echo $BAZ jake [Jan-03 11:28][~]$ [Jan-03 11:28][~]$ # And we can see that the variable is actually set: [Jan-03 11:29][~]$ echo $BAZ jake [Jan-03 11:29][~]$   What does this syntax do?  What happens to the variable that has been set?  Why does this work? $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$ Do the laws of physics evolve? Set Path for all Users (Login and Non-login Shells) Ok. So, I'm used to Ubuntu and CentOS more than redhat, so I'm hoping there's some obvious solution to this that I'm missing.  I've installed Anaconda (Python). I've placed it in:  /opt/anaconda3   I would like for myself and all users to be able to have the following in their paths:  /opt/anaconda3/bin   I have googled around and have gained a lot of conflicting information about how to "properly" accomplish my mission.  Stuff I have tried:   Adding a bash script to /etc/profile.d Editing /etc/profile Editing /etc/bashrc Adding to root bashrc Editing user .bashrc   All edits involved some form of:  PATH=/opt/anaconda3/bin:$PATH   or  export PATH=/opt/anaconda3/bin:$PATH   Option 2 yielded the addition of /opt/anaconda3/bin to the path for multiple users, but after terminal restarts still did not run the commands within bin (i.e. conda install, ipython notebook, etc.)  Any ideas? Share your experiences on exploring iOS 7 with us..  Whether you found hidden or little-known feature of iOS 7 or can explain how a big feature makes a difference, we're looking for objective reports of how the new OS affects your workflow.  Also provide details on how to achieve that feature, and if possible, include a relevant image too (hold Home and Lock for a screenshot on your iOS device).    Please post one tip per answer. Please also check to see if your answer has already been posted - duplicate answers will be deleted in favour of the original answer on any given topic. To search answers for this question use inquestion:101382 (or inquestion:this, directly from the question page) in addition to your search terms in the search box in the upper right hand corner of this page. As part of the definition of a cone spell in the handbook:     A cone's point of origin is not included in the cone's area of effect, unless you decide otherwise.   As an example, what would either case be for a casting of Burning Hands?   Could the caster include themselves? or does this mean that they can exclude the first square of the cone? Is it possible to mark the same tag as "Interesting" and "Ignored"? Can you mark the same tag as both interesting and ignored? If you can, what is the resulting appearance of a question with that tag? Is there any way to run file explorer as administrator under Windows 10 After install Windows 10, it seems that I lost the ability to run explorer.exe as administrator with a user in administrator group.   The reason I want to do that is, in our team's development environment, we are using a C# exe to do some environment configuration which will start command prompt under administrator mode and subst a drive. As a result those substed drives are not visible within the file explorer since it is run not as administrator. That will be kind of inconvenience and sometimes make mistakes.   I was able to run explorer.exe from task manager with the option "create with privileges" checked and then I can see all the drives in explorer. But now this is not working anymore.   I knew there are other options to workaround this workflow, but just want to make sure that if it is now totally impossible under Windows 10?  Any comment is appreciated. Here is a question from my number theory class.  Prove  $$(p-k)!(k-1)!\equiv (-1)^k \text{ mod p} $$ Help please! SSMS wont give up file associations The list has several empty strings in it for me right now: I know that I can update a snap package using  sudo snap refresh &lt;package&gt;   But is there a way to update all snaps, like   sudo apt dist-upgrade Physics are not simulated past a certain frame? Software to read a QR code? I saw a  reading a book, but I don't have a smart phone to check it.    So I just took a picture of it and saved my PC. And I downloaded libdecoderqr0, libdecodeqr-dev, libdecodeqr-examples for reading it. But I don't know how to use it.    What do I do next, or is there a nicer way to read QR codes? (complex measures) $d\nu=d\lambda +f\,dm \Rightarrow d|\nu|=d|\lambda| +|f|\,dm$ and $f\in L^1(\nu)\Rightarrow f\in L^1(|\nu|)$ Two simple exercises on complex measures I don't know how to solve, from Folland's Real Analysis. I find it difficult to manipulate the definitions in computations. For context, if $\nu$ is a complex measure on $(X,\mathcal{M})$ then $|\nu|$ is defined as $d|\nu|=|g|\,d\mu$ where positive measure $\mu$ and $g\in L^1(\mu)$ are such that $\nu \ll \mu$ and $\nu = g\,d\mu$ (and in the textbook it's shown that this definition does not depend on the choice of $\mu$).     1) Let $\nu$ be a reguar complex Borel measure on $\mathbb{R}^n$, $m$ denote   Lebesgue measure, and   let $d\nu = d\lambda + f\,dm$ be its Lebesgue-Radon-Nikodym   representation. Then ("it is easily verified")   $d|\nu|=d|\lambda|+|f|\,dm$.      2) If $f\in L^1(\nu)$ then $f\in L^1(|\nu|)$. I was able to prove the   reverse implication, but not this one.   The definition of $|\nu|$ in the case of real signed measures (as $|\nu|=\nu^+ + \nu^-$) makes sense to me but I can't see intuitively into the definion of $|\nu|$ for complex measures as given in the textbook. Do you have any advice on this?  Thank you! Using uniform coloring in MatrixPlot