Retrieval of square root in the field of p-adic numbers.   Find ${\sqrt 5}$ in field $\Bbb {Q_{11}}$ Find ${\sqrt 3}$ in field $\Bbb {Q_{13}}$   Any information that will help solve this task will be useful.
Suppose I want to know whether $\sqrt{7}\in\mathbb{Q}_5$, or more generally, whether $\sqrt{n}\in\mathbb{Q}_p$ for $n\in\mathbb{Z}$, $p$ an odd prime. What are the techniques for determining this? Am I supposed to "lift" it from $\mathbb{Z}/p\mathbb{Z}$ using Hensel's lemma? Could I see an example?
What is the lowest level of mathematics required in order to understand how encryption algorithms work? Consider the blow-up of the plane in one point. Let $E$ the exceptional divisor. We know that $(E,E)=-1$. What is the geometrical reason for which the auto-intersection of $E$ is $-1$? In general, what does it mean, geometrically, that a divisor has a negative self-intersection or that the "right" number of divisors gives rise to a negative intersection?  Thanks. Examples of involutions on $\mathbb{R}$ Just recently I read up something about involutions (functions $f: A \rightarrow A$ such that $f(f(x))=x$, for all $x$ in the domain of $f$), and was wondering how many (if there is a small set of general functions) such involutions exist for $A = \mathbb{R}$, or maybe $A = \mathbb{R} - S$, where S is some set of points that would make the involution work if they were left out of $A$. In general I'm interested in real functions that are involutions, continuous or otherwise.  There were some examples I found , but any more would be certainly very interesting! How do I get the first element from an IEnumerable in .net? I often want to grab the first element of an IEnumerable&lt;T&gt; in .net, and I haven't found a nice way to do it.  The best I've come up with is:  foreach(Elem e in enumerable) {   // do something with e   break; }   Yuck!  So, is there a nice way to do this? Ubuntu gets stuck in a login loop My Ubuntu is stuck in a login loop when trying to enter my desktop. When I login, the screen gets black and soon after that the login screen comes back.   I've read that the problem might be caused by an error depending on the graphics, here's my graphics card: ATI Radeon 7670M find the value of limit Find the value of $$ \lim_{n\to\infty} \sum_{k=n+1}^{2n} \frac{1}{k} $$ I think it become a series with each denominator approach infinity. so the value of the limit is 0. am i true ? Partitions for Ubuntu and Windows 7 dual boot on SSD + HDD setup Negative solution to $x^2=2^x$ Just out of curiosity I was trying to solve the equation $x^2=2^x$, initially I thought there would be just the two solutions $x=2$ and $x=4$, but wolfram shows that the two equations intersect at not 2 but 3 locations, the third being a negative value of $x$. The third solution isn't obvious like the other two, so I just have a few questions about the negative solution. Is it rational? is it commonly represented with a greek letter? If it is irrational is there a way to approximate it? What is the difference between a synchronized method and synchronized block in Java? Recently, Firefox has changed the default folder for newly created bookmarks (using Ctrl+D) from the Bookmarks Menu to "Other Bookmarks".  Is there some way to change that back? Or some alternate method instead of Ctrl+D?  EDIT  A native way to do this (without an add-on) would be best. Converting a DC motor to a generator Can ANY DC motor become a generator if torque is applied to the shaft? example:   And by any, I mean the really cheap ones too.  Or is there a specific type of motor (of which I do not know the name) that can work in both directions (battery to power the shaft OR torque-on-shaft to create a voltage difference between the terminals). Compute $\sum_{n=1}^\infty \frac{1}{(n+1)\sqrt[]{n}+n\sqrt[]{n+1}}$ Compute   $\sum_{n=1}^\infty \frac{1}{(n+1)\sqrt[]{n}+n\sqrt[]{n+1}}$ I'm trying to use the ant migration tool to retrieve all the Custom Metadata records for a single Custom Metadata Type.  However, I can't figure out the syntax.  I would think that something like this would work:    &lt;Package xmlns="http://soap.sforce.com/2006/04/metadata"&gt;     &lt;types&gt;         &lt;members&gt;State_Sales_Tax.*&lt;/members&gt;         &lt;name&gt;CustomMetadata&lt;/name&gt;     &lt;/types&gt;     &lt;types&gt;         &lt;members&gt;State_Sales_Tax__mdt&lt;/members&gt;         &lt;name&gt;CustomObject&lt;/name&gt;     &lt;/types&gt;     &lt;version&gt;36.0&lt;/version&gt; &lt;/Package&gt;   but no dice. How do I increase the open files limit for a non-root user? This is happening on Ubuntu Release 12.04 (precise) 64-bit Kernel Linux 3.2.0-25-virtual  I'm trying to increase the number of open files allowed for a user. This is for my eclipse java application where the current limit of 1024 is not enough.  According to the posts I've found so far, I should be able to put lines into   /etc/security/limits.conf like this:  soft nofile 4096 hard nofile 4096   to increase the number of open files allowed for all users.  But that's not working for me and I think the problem is not related to that file.  For all users, the default limit is 1024, regardless of what is in /etc/security/limits.conf (I rebooted after changing that file)  $ ulimit -n 1024   Now, despite the entries in /etc/security/limits.conf I can't increase that:  $ ulimit -n 2048   -bash: ulimit: open files: cannot modify limit: Operation not permitted The weird part is that I can change the limit downwards, but can't change it upwards - even to go back to a number which is below the original limit:  $ ulimit -n 800 $ ulimit -n 800  $ ulimit -n 900  -bash: ulimit: open files: cannot modify limit: Operation not permitted   As root, I can change that limit to whatever I want, up or down. It doesn't even seem to care about the supposedly system-wide limit in /proc/sys/fs/file-max  # cat /proc/sys/fs/file-max 188897  # ulimit -n 188898 # ulimit -n  188898   But even I get eclipse to run as root, my application still crashes because of "Too Many Open File" exception!   So far, I haven't found any way to increase the open files limit for a non-root user.  How should I properly do this? I have looked at several other posts but no luck! Is it advisable to use lead-free PCB and solder paste on a first SMD assembly job? How hard do early-career academics in the United States work, really? I'm finishing a PhD in Sweden this year and seeking a post-doc in the United States (atmospheric remote sensing).  It is clear that a job in academia is not a nine to five job; nor do I want it to be.  I often work late in the evening when I'm on a productive spur.  When an important dead-line is coming up, one needs to work harder, and there is no overtime paid.  I accept that.  However, no matter how much I'm interested in my research, I do enjoy and need a reasonable amount of spare time, too; relaxing on the weekends, occasionally a long weekend away, and sometimes a longer trip, such as three weeks in the wilderness.  Regarding the normal work ethos for early-career academics in the United States, I have a hard time judging what is normal and what is excessive.  Some examples:    warning post-docs that he expects all of the members of the group to work evenings and weekends. My apparent naïveté in believing that , despite Wikipedia  as a rest from work, or a hiatus, often lasting from two months to a year.  I was thinking of my friend, who spent a year between his PhD and his first post-doc travelling from France to Mongolia mostly on foot. Someone  I am not a good role model, but when I don't work on a Sunday, I count that as a vacation day.  having no employment-related benefits such as paid vacations, sick leave, or unemployment compensation. Question    We don't have such a work-ethos where I'm at.  I belief that working too hard risks stress and burn-out, and does not increase productivity in the long run, nor human well-being.  I want to do science.  Doing science makes me happy, but having time to relax while not doing science is important for me.  Does the selection of examples I gave above represent a normal situation in U.S. early-career academia?  Should I expect an attitude where asking for a 3-week holiday during the summer is considered as being not serious, or is the situation in practice usually not as bad as the examples above make it sound?  How hard to early-career academics in the United States work, really? How do I find out which version and derivative of Ubuntu is right for my hardware in terms of minimal system requirements? For a given hardware configuration, how do I find out if Ubuntu will run on it? What considerations should I take into account when choosing an Ubuntu version and  such as:    with a lighter desktop than the usual Gnome and Unity  with the even lighter LXDE desktop   Obviously Ubuntu does not run on some processor architectures. So how do I go about choosing the right version and derivate. How can I find out the minmal system requirements? 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? I want to install OpenOffice. What can I do to install it on my computer? $f$ is entire and maps a rectangle to a rectangle, then $f$ is linear The problem is in the subject line, I have it for homework. $f$ is a complex valued function. For completeness:     Prove that if $f$ is an entire function and for some rectangle $R$, the image $f(R)$ is also a rectangle, then $f$ is linear.    The composition of linear maps is linear, so we can choose the two rectangles to have two edges coinciding with the real and imaginary axes, and their common vertex at the origin. So as a portion of the real line gets mapped to itself, we can take $f$ to be the analytic continuation of a real function. Then I'm stuck.  This question is confusing me. It's well known from the Riemann mapping theorem that there exists many holomorphic functions taking any rectangle to any other given rectangle. But if $f$ is linear, then the two rectangles must be similar. So the added restraint of $f$ being entire seems to be causing this hassle. Thoughts?