Let $k$ be a field and let $R$ be the subring of $k[x]$ consisting of all polynomials having no linear term, that is, $f (x) \in R$ if and only if $f (x) = s_0 +s_2x^2 +s_3x^3 +\ldots + s_nx^n$.   Show that $x^5$ and $x^6$ have no gcd in $R$.
Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$.   I am reasoning for the absurd and I am assuming that $gcd (x ^ 5, x ^ 6) = ax ^ 2 + bx ^ 3\in\mathbb{Q}[x^2 ,x^3 ] $, then, for the Bezout identity there exists $p (x) = cx ^ 2 + dx^3$ and  $q(x)= ex ^ 2 + fx ^ 3$ in $\mathbb{Q}[x^2 ,x^3 ]$ such that $x ^ 5p (x) + x ^ 6q (x) = ax ^ 2 + bx ^ 3$, which is absurd since $$9 = deg (x ^ 5p (x) + x ^ 6q (x)) \neq  deg (ax ^ 2 + bx ^ 3) = 3.$$  Is it okay what I did? Thank you very much.
I often hear people saying that "I am" is the shortest sentence in the English language.  I know that there are also discussions about sentences using the imperative mood such as "Go." that would be shorter, but my question is this:  Why would we (the people saying "I am" is a full sentence) not accept "I'm" as a complete sentence?  Is there some unwritten rule about contractions that says "I'm" wouldn't be correct? bash - problems reading text file and printing line by line How to move a visible file to a hidden folder? I've only been using Ubuntu for two weeks and have been able to find answers to any problem I had, however this one defies my searches.  I want to move a visible programme file (wallch) to a hidden directory ./config/autostart.  I have been able to enter this directory and the subject file is not there and I want to have this program start at bootup.  Can anyone assist?  Thanking you How to loop the particle simulation indefinitely? I've made a simple snowfall particle simulation (using gandalf3's solution: ). Is there a way to loop the animation, so the transition between last frame and first frame is not visible? Is there a setting to do it? Testing equality of coefficients from two different regressions This seems to be a basic issue, but I just realized that I actually don't know how to test equality of coefficients from two different regressions. Can anyone shed some light on this?   More formally, suppose I ran the following two regressions:  $$ y_1 = X_1\beta_1 + \epsilon_1 $$ and  $$ y_2 = X_2\beta_2 + \epsilon_2 $$ where $X_i$ refers to the design matrix of regression $i$, and $\beta_i$ to the vector of coefficients in regression $i$. Note that $X_1$ and $X_2$ are potentially very different, with different dimensions etc. I am interested in for instance whether or not $\hat\beta_{11} \neq \hat\beta_{21}$.   If these came from the same regression, this would be trivial. But since they come from different ones, I am not quite sure how to do it. Does anyone have an idea or can give me some pointers?   My problem in detail: My first intuition was to look at the confidence intervals, and if they overlap, then I would say they are essentially the same. This procedure does not come with the correct size of the test, though (i.e. each individual confidence interval has $\alpha=0.05$, say, but looking at them jointly will not have the same probability). My "second" intuition was to conduct a normal t-test. That is, take   $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11})} $$  where $\beta_{21}$ is taken as the value of my null hypothesis. This does not take into account the estimation uncertainty of $\beta_{21}$, though, and the answer may depend on the order of the regressions (which one I call 1 and 2).   My third idea was to do it as in a standard test for equality of two coefficients from the same regression, that is take  $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11}-\beta_{21})} $$   The complication arises due to the fact that both come from different regressions. Note that   $$ Var(\beta_{11}-\beta_{21}) = Var(\beta_{11}) + Var(\beta_{21}) -2 Cov(\beta_{11},\beta_{21}) $$ but since they are from different regressions, how would I get $Cov(\beta_{11},\beta_{21})$?  This led me to ask this question here. This must be a standard procedure / standard test, but I cound not find anything that was sufficiently similar to this problem. So, if anyone can point me to the correct procedure, I would be very grateful! I am solving the TOEIC workbook, by the way, I faced conflict that doesn't match meaning I knew.  The problem is     The newest branch of the library is located in/on Pine Street.   I think either 'in' or 'on' have the same meaning. But the commentary said that 'be located in' is mainly writing in front of building name and city name.   I don't know what is correct? I am trying to run a simple echo script via crontab. I set that to run every minute, but it doesn't give output on shell screen. However, it runs fine when I run the script independently.   Script  #!/bin/bash echo "Test Script"   Crontab entry:  root@example-server ~]# cat /etc/crontab SHELL=/bin/bash PATH=/sbin:/bin:/usr/sbin:/usr/bin MAILTO=root HOME=/     # For details see man 4 crontabs  # Example of job definition: # .---------------- minute (0 - 59) # |  .------------- hour (0 - 23) # |  |  .---------- day of month (1 - 31) # |  |  |  .------- month (1 - 12) OR jan,feb,mar,apr ... # |  |  |  |  .---- day of week (0 - 6) (Sunday=0 or 7) OR sun,mon,tue,wed,thu,fri,sat # |  |  |  |  | # *  *  *  *  * user-name command to be executed   *  *  *  *  * root /root/test.sh In Europe usually you don't need to go through immigration to change planes (unless you are entering/exiting Schengen). Is it the case in the US as well? May I catch a connecting flight on one of the US airports if I don't have US visa? Is it possible to get the information for a device tree using /sys of a running kernel? Commonly for arm systems, device trees supply hardware information to the kernel (Linux). These device trees exist as dts (device tree source) files that are compiled and loaded to the kernel. Problem is that I do not have access to such a dts file, not even to a dtb file.   I have access to /sys and /proc on the machine and I wanted to ask if that would allow me to "guess the correct values" to be used in a dts?  Also potential answer could highlight additionally the aspect if the answer to this question also depends on whether the device tree interface was used in the first place (i.e. a dtb was created and provided to the kernel) instead of some more hacking "we simply divert from vanilla and patch the kernel so as to solve the device information problem for our kernel only"-solution? Find the last $2$ digits of $$\large7^{7^{7^{10217}}}$$   So far I have: $17\cdot 601=10217$ and $7=7 \pmod{10}$. Any help greatly appreciated. How to customize Windows 10 "6 Folders" in My Computer I would like to customize the folders in the Default Folders group of My computer. It is not necessary to have the same folders appear on the left sidebar and in the middle view. How can I do that ? Is there an easy way to attach source in Eclipse? I'm a big fan of the way Visual Studio will give you the comment documentation / parameter names when completing code that you have written and ALSO code that you are referencing (various libraries/assemblies).  Is there an easy way to get inline javadoc/parameter names in Eclipse when doing code complete or hovering over methods? Via plugin? Via some setting? It's extremely annoying to use a lot of libraries (as happens often in Java) and then have to go to the website or local javadoc location to lookup information when you have it in the source jars right there! How does "curved space" explain gravitational attraction? They say that gravity is technically not a real force and that it's caused by objects traveling a straight path through curved space, and that space becomes curved by mass, giving the illusion of a force of gravity.  That makes perfect sense for planetary orbits, but a lot less sense for the expression of gravity that we are the most familiar with in our day-to-day lives: "what goes up must come down."  Imagine that I hold a ball in my hand, several feet off the ground, with my fingers curled around it, and my hand is above the ball.  Then I open my fingers, releasing my grip on it, being very careful to not impart any momentum to the ball from my hand as I do so.  An object at rest remains at rest unless acted upon by an outside force.  If the ball is not moving (relative to my inertial reference frame), it has no path to travel that's any different from the Earth's path through space.  It should remain at rest, hanging there in the air.  And yet it falls, demonstrating that an outside force (gravity) did indeed act upon it.  How does curved space explain this? Today I open my SO account like I always do. But the UI of menus has changed. I am not able to see the chat tab, the logout link or the privileges link. Are they missing or moved somewhere else?  Changed UI: How do I boot from ISO file stored on hard disk? How does gravity escape a black hole? My understanding is that light can not escape from within a black hole (within the event horizon).  I've also heard that information cannot propagate faster than the speed of light.  It would seem to me that the gravitational attraction caused by a black hole carries information about the amount of mass within the black hole.  So, how does this information escape?  Looking at it from a particle point of view: do the gravitons (should they exist) travel faster than the photons? In high school we learned to say "than I" and "as I" because you could potentially add an "am" to the end of the sentence.  Examples:  "She is smarter than I."  (Think: "...than I am.") "He is as tall as I."  (Think "...as I am.")  So analogously, shouldn't it be "like I" as well:  "He is sincere, just like I."  (Think: "... like I am.")  But universally, it seems that we use "like me".  Where does this reasoning break down?  Is there history here? Can I have two Schengen visas in one passport? Positive euphemism for "harped on" How can I replace harped on with a more positive expression?  The production manager harped on the new quality assurance regulations for nearly an hour. During the next quarter at uni, I'll be taking a course in real analysis and since I prefer studying with an additional text I thought I'd come here to look for some book recommendations.  My background: I'm comfortable with linear algebra and single variable calculus, but shakey on multivariable calculus. I did have a slight introduction in to geometry.  For linear algebra I found the common recommendation of Hoffman &amp; Kunze to be very helpful (although the abstractness came as a bit of a kicker at first).   For Single variable calculus I used the book Calculus by Adams and Essex, which I found frustrating to work with, because it often left me guessing at what they were trying to do or why they were doing it. I also felt that they exercised a certain lack of rigour. (Which may be due to the subject which would make it hard to derive from axioms.)  Details: Though it would be a complementary text, I'd prefer one which will still hold value as a work of reference at a later point in my studies. I've heard that Rudin's Principles of mathematical analysis is one of the better textst out there, but also read several comments discouraging Rudin's book as a first introduction to real analysis, hence my quest to gather more information and maybe get some more personalised recommendations.  The course: The mandatory course literature will consist of lecture notes, but 4 texts are suggested as recommended reading.   T.M. Apostol, Mathematical analysis. Addison-Wesley (1974) J. Dieudonné, Foundations of Modern Analysis. Academic Press (1960) A. van Rooij, Analyse voor Beginners.Epsilon Uitgaven, no. 6 (2003)   R.S. Strichartz, The way of analysis (1995)   Thanks in advance!